Question: Calculate the walking time in minutes for a distance of 9 km at a speed of 3.0 km/h, then add the fixed break time of 24 minutes to find the total duration in minutes.


Question: Calculate the total duration in minutes for a 9-kilometer walk at a speed of 3 kilometers per hour, including 24 minutes spent at a coffee shop.


Question: Apply the logarithmic power rule $\log_a(b^c) = c \log_a b$ to the first equation $\log_x(y^x) = 10$ and express the result in terms of $x$ and $\log_x y$.


Question: Given distance 9 km, total time 4 hours, and speed 2.5 km/h, calculate the constant coffee break time k in hours spent in the coffee shop.


Question: A walk of 9 km at a speed of 2.5 km/h takes 4 hours total (including the coffee shop break time $t$). Calculate the value of $t$ in minutes.


Question: Using the fact that $\log_x y = \frac{10}{x}$ and $\log_x y = \frac{2y}{5}$, solve for $y$ in terms of $x$ by equating the two expressions.


Question: Given the equations $\log_x\left(y^x\right)=10$ and $\log_y\left(x^{4y}\right)=10$, apply the logarithm power rule $\log_b(a^c) = c \log_b a$ to simplify the left-hand side of each equation. What are the resulting simplified equations?


Question: Given distance 9 km, walking speed 3 km/h, and constant stop time 24 minutes, calculate the total time in minutes.


Question: In Scenario 1, the walking distance is 9 km at a speed of 2.5 km/h, resulting in a total time of 4 hours which includes $t$ minutes of coffee break. How many minutes does the coffee break last?


Question: Calculate the positive value of s in km/h that satisfies the quadratic equation: 4s^2 + 8s - 45 = 0, derived from 9/s - 9/(s+2) = 1.6.


Question: Given a distance of 9 km, a walking speed of 3 km/h, and a constant break time of 24 minutes, what is the total time in minutes?


Question: Using the isolated values $\log_x y = \frac{10}{x}$ and $\log_y x = \frac{5}{2y}$, apply the identity $\log_x y \cdot \log_y x = 1$ to solve for $xy$.


Question: Given a distance of 9 km traveled at a constant speed of 3 km/h, calculate the duration spent walking in minutes.


Question: With a 9 km walk, if the total time including a fixed coffee break was 4 hours at 2.5 km/h, and the new speed is 3 km/h, find the new total time in minutes including the same coffee break.


Question: Aya walks 9 km. In the first case, her speed was 2.5 km/h and the total time was 4 hours, including $t$ minutes of coffee. How long is $t$ in minutes?


Question: Solve the equation $9/s - 9/(s+2) = 1.6$ for the positive value of $s$ (in km/h).


Question: Solve the equation $\frac{9}{s} - \frac{9}{s+2} = 1.6$ for the positive value of $s$. Return the numerical value of $s$.


Question: Solve for the positive real number $s$ in the equation $\frac{9}{s} - \frac{9}{s+2} = 1.6$.


Question: Determine the exponential equations equivalent to $\log_x(y^x)=10$ and $\log_y(x^{4y})=10$.

\begin{aligned} x^{10} &= y^x \\ y^{10} &= x^{4y} \end{aligned}


Question: Calculate the value of $s$ (in km/h) by solving the equation $\frac{9}{s} - \frac{9}{s+2} = 1.6$ derived from the two walk scenarios described above (9 km distance, 4 hours vs 2.4 hours including constant break time).


Question: Clean up decimals: 4 hours = 240 minutes. At speed s = 2.5 km/h for 9 km, calculate coffee break time t in minutes (Total time minus walking time). Then, for the new speed s + 0.5 = 3.0 km/h, calculate total time in minutes (New walking time + t).


Question: Aya walks 9 km. Trip 1 takes 4 hours at speed $s$ km/h. Trip 2 takes 2.4 hours at speed $s+2$ km/h. Both times include a constant stop. Using Walking Time = Distance/Speed, derive and solve the equation $4 - 2.4 = \frac{9}{s} - \frac{9}{s+2}$ for the positive real number $s$.


Question:
Based on the equations $x \log_x y = 10$ and $\log_y(x^{4y}) = 10$:
1. Apply the power rule to the second equation to obtain $4y \log_y x = 10$.
2. Isolate $\log_x y = 10/x$ and $\log_y x = 10/(4y)$.
3. Use the identity $\log_x y \cdot \log_y x = 1$ to substitute these values.
4. Solve the resulting algebraic equation for $xy$.
Provide the final calculated value of $xy$.


Question: Using the equation $\log_x(y^x)=10$, determine the expression for $\log_x y$ in terms of $x$.


Question: Compute $115 + 256$.


Question: Considering the equation $\log_x(y^x)=10$, apply the logarithmic power rule $\log_b(a^c)=c \log_b a$ to simplify it. Let $t = \log_x y$. Express $x$ in terms of $t$.


Question: What is $11+128$?


Question: Using the power property of logarithms ($\log_b(M^n) = n \log_b M$), simplify the equation $\log_y(x^{4y}) = 10$ to solve for $\log_y x$.


Question: Solve the equation $\frac{9}{s} - \frac{9}{s+2} = 1.6$ for the value of $s$.


Question: In the equation $\log_x(y^x) = 10$, simplify the left-hand side using the power property of logarithms to find the value of $\log_x y$.


Question: Given $\log_x y = \frac{10}{x}$ and $\log_y x = \frac{5}{2y}$, find the value of the product $xy$ using the identity $\log_x y \cdot \log_y x = 1$.


Question: Aya walks 9 km. In the first scenario, walking at speed $s$ km/h and resting for $t$ minutes takes 4 hours total. In the second scenario, walking at speed $s+2$ km/h and resting for $t$ minutes takes 2 hours 24 minutes total. Find $s$.


Question: Calculate the sum of 1 and 115.


Question: Calculate the sum $90 + 24 + 1$.


Question: Calculate the value of the expression \binom{4}{3} \times \binom{6}{1}.


Question: Calculate the numerical value of $\binom{4}{3} \times \binom{6}{1}$.


Question: Calculate the sum of 163 and 256.


Question: In a lottery of 10 numbers, if 4 are drawn and Jen picks 4, calculate the number of ways Jen picks 4 numbers that are exactly equal to the 4 drawn numbers. This is $\binom{4}{4} \times \binom{6}{0}$.


Question: Calculate the value of $\binom{4}{2} \times \binom{6}{2}$.


Question: Calculate the numerical value of $\binom{4}{2} \times \binom{6}{2}$.


Question: Calculate the number of positive integer solutions to $x+y+z=8$. Let this be $N_3$. Calculate the number of positive integer solutions to $u+v=8$. Let this be $N_2$. Compute the final answer $Ans = 2 \times N_3 \times N_2$ and report it.


Question: Given $AD = \frac{325}{22}$ and $BD = \frac{225}{22}$, calculate $AP = AD - \frac{BD^2}{AD}$ and simplify to $\frac{m}{n}$.


Question: Solve the system of equations $x^2 + y^2 = 41$, $x^2 + z^2 = 80$, and $y^2 + z^2 = 89$ for positive real numbers $x, y, z$.


Question: Calculate the numeric value of V = 160/3. Also calculate the numeric value of S = 2 * sqrt(25*16 + 16*64 + 64*25).


Question: Given $\log_x y = \frac{10}{x}$ and the equation $4y \log_y x = 10$, what is the numerical value of the product $xy$?


Question: Given $x = \frac{10}{t}$ where $t = \log_x y$, simplify the equation $\log_y(x^{4y}) = 10$ to find $y$ in terms of $t$.


Question: Find positive real numbers $x, y, z$ that satisfy the system of equations $x^2 + y^2 = 41$, $y^2 + z^2 = 80$, and $z^2 + x^2 = 89$.


Question: Celebrate the end of the journey!


Question: Calculate the value of the binomial coefficient $\binom{7}{2}$.


Question: Calculate the sum of binomial coefficients $\binom{8}{0} + \binom{8}{1} + \binom{8}{2} + \binom{8}{3} + \binom{8}{4}$.


Question: What is the value of the binomial coefficient $\binom{7}{1}$?


Question: How many ways are there to choose 4 distinct numbers from a set of 10?


Question: Count the number of positive integers $n$ such that $n \le 2024$ and $n \equiv 0 \pmod 5$.


Question: Calculate $21 \times 7 + 21 \times 7$.


Question:
Multiply the two equations given below:
Equation A: $x \log_x y = 10$
Equation B: $4y \log_y x = 10$
Combine LHS and RHS. Using the identity $\log_x y \cdot \log_y x = 1$, simplify the expression to solve for the product $xy$.


Question: Calculate the number of ways to choose 4 numbers from 10 such that the chosen set intersects a fixed set of 4 numbers in at least 2 elements.


Question: In the set of positive integers $n$ less than or equal to 2024, how many satisfy the condition that $n$ leaves a remainder of 2 when divided by 5?


Question: What is the result of multiplying 2, 21, and 7?


Question: Calculate the inradius $r$ given $V = 160/3$ and $S = 24\sqrt{21}$. Express $r$ in the form $\frac{m \sqrt n}{p}$ where $m$ and $p$ are relatively prime and $n$ is square-free. Find the value of $m+n+p$.


Question: Calculate the numerical value of the binomial coefficient \binom{7}{2} (7 choose 2).


Question: Count the number of positive integers $n$ less than or equal to 2024 such that $n$ leaves a remainder of 0 or 2 when divided by 5.


Question: Calculate the number of ways to choose 4 numbers from a set of 10 that share exactly 2, 3, or 4 numbers with a specific target set of 4 numbers. Evaluate the sum: $\binom{4}{2}\binom{6}{2} + \binom{4}{3}\binom{6}{1} + \binom{4}{4}\binom{6}{0}$.


Question:
The probability is (Good Subsets) / 256.
Calculate Good Subsets = Subsets of size 0, 1, 2, 3 + Subsets of size 4 with non-full difference set.
Size 0: 1.
Size 1: 8.
Size 2: 28.
Size 3: 56.
Total size <=3: 93.
Bad subsets of size 4: 24 (based on cyclic gap classifications).
So Good size 4: 70 - 24 = 46.
Total Good = 139.
Fraction 139/256.
Simplify and sum m+n.


Question: Given $\log_x y = \frac{10}{x}$ and the equation $\log_y(x^{4y}) = 10$, calculate the value of $xy$.


Question: Calculate the number of ways to choose exactly 2 numbers from 4 specific numbers multiplied by the number of ways to choose the remaining 2 numbers from the other 6 numbers.


Question: Calculate the value of $4 \times 75 + \frac{96}{4}$.


Question: Solve for $x^2, y^2, z^2$ using the equations $x^2+y^2=41$, $y^2+z^2=80$, and $z^2+x^2=89$.


Question: In a lottery with a set of 10 distinct numbers, Jen chooses 4 distinct numbers. What is the number of possible distinct sets of numbers Jen could have chosen?


Question: Calculate the number of ways to choose exactly 3 numbers matching from Jen's 4 choices and 1 number not matching from the remaining 6. Express this as C(4,3) * C(6,1).


Question: Solve for the positive real number $x$ satisfying $x^3 - 27x + 46 = 0$.


Question: Given real numbers $x, y > 1$ satisfying $\log_x(y^x)=10$, compute the value of $\log_x y$.


Question: Calculate the value of $K_2 = -4 \times 117 + \frac{144}{4}$.


Question: In a rectangular box with dimensions $L, W, H$, the face diagonals have squared lengths $L^2+W^2=41$, $L^2+H^2=80$, and $W^2+H^2=89$. Find the value of $LWH$.


Question: How many positive integers $n$ satisfy $1 \le n \le 2024$ and ($n \equiv 0 \pmod 5$ or $n \equiv 2 \pmod 5$)?


Question: Let x and y be two dimensions of a box such that x=y. Given the conditions xy+yz+zx=27 and xyz=23, find the positive real solution for x and the corresponding z. Then calculate the value of x^2 + y^2 + z^2.


Question: In the given system of equations $\log_x(y^x)=10$ and $\log_y(x^{4y})=10$ with $x, y > 1$, apply the power rule $\log_b(a^k) = k \log_b a$ to rewrite these equations in the forms $C_1 \log_x y = C_2$ and $C_3 \log_y x = C_4$. Find the constants $C_i$.


Question: Calculate the sum of 90, 24, and 1.


Question: Calculate the number of ways to choose 4 distinct numbers from a set of 4 specific numbers.


Question: Define $h(t) = 4 \left| \left| t - \frac{1}{2} \right| - \frac{1}{4} \right|$ for $t \in [0, 1]$. Identify the critical points where the expression changes slope and provide the simplified linear expression for $h(t)$ on the intervals $I_1 = [0, \frac{1}{4}]$, $I_2 = [\frac{1}{4}, \frac{1}{2}]$, $I_3 = [\frac{1}{2}, \frac{3}{4}]$, and $I_4 = [\frac{3}{4}, 1]$.


Question: Given a tetrahedron with Volume V = 160/3 and Total Surface Area S = 24\sqrt{21}, calculate the inradius r using the relation V = (1/3)Sr. Express r in the form \frac{m \sqrt n}{p} where m, n, p are positive integers, m and p are relatively prime, and n is square-free. Then find the value of m+n+p.


Question: Calculate the number of ways to choose exactly 2 numbers from the 4 winning numbers and exactly 2 numbers from the 6 non-winning numbers.


Question: Calculate the sum of 90, 24, and 1.


Question: Calculate the number of positive integer solutions to the equation $u_1 + u_2 = 8$.


Question: In a scenario where Jen chose 4 specific numbers and 6 remain unchosen, calculate the number of ways to pick 4 numbers such that exactly 3 are from Jen's 4 choices and 1 is from the 6 others.


Question: What is 21 multiplied by 7 plus 21 multiplied by 7?


Question: What is the sum of 1 and 115?


Question: In this game, a position is Losing if all reachable positions from it are Winning; otherwise it is Winning. Consider the game where a player removes 1 or 4 tokens. For $n=0$ through $n=20$, determine whether the position is Winning (W) or Losing (L). Based on these values, state the general condition for an integer $n$ to be a Losing position (preferably in terms of modulo arithmetic).

n \equiv 0 \pmod 5 \text{ or } n \equiv 2 \pmod 5


Question: In a scenario with 5 segments of directions R, U, R, U, R, the number of Right steps is distributed among 3 segments (the 1st, 3rd, and 5th). Given there are exactly 8 Right steps in total and each segment must contain at least 1 step, find the number of integer solutions to $r_1 + r_2 + r_3 = 8$ where $r_i \ge 1$.


Question: Based on the established P-position pattern ($n \equiv 0 \text{ or } 2 \pmod 5$), calculate the number of positive integers $n$ such that $1 \le n \le 2024$ and $n$ is a P-position. Express the final count clearly.


Question: Verify the count of good sets of size 4. Specifically, confirm that IS sets for k=1 (2 sets), k=2 (4 sets), and k=4 (16 sets) are mutually exclusive.


Question: Calculate the area of a triangle with side lengths $\sqrt{41}$, $\sqrt{80}$, and $\sqrt{89}$. Express the answer in simplest radical form.


Question: Calculate the following values: N(4) = C(4,4) * C(6,0); N(3) = C(4,3) * C(6,1); N(2) = C(4,2) * C(6,2).


Question: Calculate the inradius $r$ of a tetrahedron inscribed in a rectangular box of dimensions $5, 4, 8$. The volume is given by $V = \frac{5 \times 4 \times 8}{3}$. The faces are congruent triangles with side lengths $\sqrt{41}, \sqrt{80}, \sqrt{89}$. Use Heron's formula to find the area of one face, calculate the total surface area $S = 4 \times \text{Area}_{\text{face}}$, and then find $r = \frac{3V}{S}$. Express the result in the form $\frac{m\sqrt{n}}{p}$ where $m, p$ are coprime and $n$ is square-free.


Question: Given $x=5/2$, $y=2$, $z=4$. Using $V = \frac{8}{3}xyz$ and $S = 8\sqrt{x^2y^2+y^2z^2+z^2x^2}$, calculate the inradius $r = \frac{3V}{S}$. Express $r$ in the form $\frac{m\sqrt{n}}{p}$ where $m,n,p$ are positive integers, $m,p$ coprime, $n$ square-free. Then calculate $m+n+p$ and report the value.


Question: Solve the system of equations $x^2+y^2=41, x^2+z^2=80, y^2+z^2=89$ for $x^2, y^2, z^2$.


Question: Calculate $u = 4 \times (75 + 117i)$. Return its real part ($u_r$) and imaginary part ($u_i$).


Question: In the expression $(75+117i)z+\frac{96+144i}{z}$ with $|z|=4$, let $z=4e^{i\theta}$. What is the result of computing $(75+117i) \times 4$?


Question: Verify that m = 110 satisfies 110^4 + 1 is divisible by 17^2 = 289, and confirm no smaller positive integer m satisfies the condition for p = 17.


Question: Calculate the real part $v_r$ and imaginary part $v_i$ of the complex number $v = \frac{96+144i}{4}$.


Question: Calculate the complex numbers A = 4(75+117i) and B = (96+144i)/4. Provide them in the form x+yi.


Question: What is the value of $\alpha = 4 \times (75 + 117i)$?


Question: Calculate the value of $\beta = (96+144i) / 4$.


Question: Calculate sqrt((300+24)^2 + (36-468)^2).


Question: Calculate \sqrt{324^2 + 432^2}


Question: For u in [-1, 1], what is the explicit piecewise linear formula for h(u) = 4 * g(f(u)) where f(x) = ||x| - 1/2| and g(x) = ||x| - 1/4|?


Question: Calculate the sum of the counts of the four sets of residents: |D| + |G| + |S| + |C|. The values are 195, 367, 562, and 900 respectively.


Question: What is the value of the square root of 291,600?


Question:
Solve for the number of residents with exactly 4 items (let this be x).
Constraints derived from Facts:
1. Residents(1) + x = 229
2. Residents(1) + 4*x = 448
(Calculation Note: 229 = 900 - (437 + 234); 448 = 2024 - (2*437 + 3*234)).


Question: How many integers $n$ exist such that $1 \le n \le 2024$ and ($n \equiv 0 \pmod 5$ or $n \equiv 2 \pmod 5$)?


Question: Calculate the sum of the counts for the four categories: Diamond Ring (195), Golf Clubs (367), Garden Spade (562), and Candy Hearts (900).


Question: Given $p=17$, verify that $m=110$ is a solution to $m^4 \equiv -1 \pmod{17^2}$ and check if any smaller positive integer $m$ exists by verifying if the residues of 110, 151, 155, 179 are the unique solutions in [1, 289]. Specifically confirm 110 is indeed the global minimum.


Question: Given M = 324 and N = -432, calculate the value of sqrt(M^2 + N^2).


Question: Solve the expression $R = \frac{192}{5}$ to confirm $m=192, n=5$ and $m+n$.


Question:
We have 900 residents. Each resident owns at least 1 item (Candy Hearts). We know:
- Number of residents owning exactly 2 items = 437
- Number of residents owning exactly 3 items = 234
- Sum of all individual item ownership counts across all sets = 2024
Let $x$ be the number of residents owning exactly 4 items, and $y$ be the number owning exactly 1 item.
Set up the following system of equations:
1. $y + 437 + 234 + x = 900$ (Total people)
2. $1 \cdot y + 2 \cdot 437 + 3 \cdot 234 + 4 \cdot x = 2024$ (Total ownership counts)
Solve this system for $x$.


Question: Calculate the product of $\binom{4}{2}$ and $\binom{6}{2}$.


Question: Determine the Winning/Losing status (W/L) for the positions $n=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. Specifically, indicate which of these values are Losing positions (where the current player has no winning move).


Question: Calculate the value of $(\text{Re}(\alpha)+\text{Re}(\beta))^2+(\text{Im}(\beta)-\text{Im}(\alpha))^2$ where $\alpha = 300+468i$ and $\beta = 24+36i$.


Question: Calculate the number of integers $n$ such that $1 \le n \le 2024$ and ($n \equiv 0 \pmod 5$ or $n \equiv 2 \pmod 5$).


Question: Using the stars and bars formula, calculate the number of compositions of 8 into exactly 3 positive parts (i.e., $\binom{8-1}{3-1}$).


Question: Evaluate the arithmetic expression $75 \times 4 + \frac{96}{4}$.


Question: Calculate the number of compositions of 8 into exactly 2 positive integers using the Stars and Bars method $\binom{n-1}{k-1}$. Provide the result in \boxed{}.


Question: What is the value of $2 \times 21 \times 7$?


Question: Determine the infimum of the function $f(t) = \frac{480}{11t - 5}$ for $t$ in the open interval $(5/11, 6/11)$.


Question: In a triangle with circumradius R=13 and inradius r=6, what is the value of the squared distance between the circumcenter and the incenter (Euler's theorem)?


Question: Calculate the value of $-117 \times 4 + \frac{144}{4}$.


Question: In the subtraction game where a player can remove 1 or 4 tokens, determine if positions $n=0$ to $n=15$ are P-positions or N-positions. A position is P if all valid next moves lead to N; otherwise N. List the status for each $n$.

P, N, P, N, N, P, N, P, N, N, P, N, P, N, N, P


Question: Calculate the product of 21 and 7.


Question: What is the number of ways to partition the integer 8 into exactly 3 positive integers?


Question: What is 147 added to itself (multiply 147 by 2)?


Question: What is the value of $\sqrt{324^2 + (-432)^2}$?


Question: Given the function $f(t) = \frac{480(1+t)}{6-5t}$ defined for $t \in (5/6, 6/5)$, find the greatest real number that is less than $f(t)$ for all $t$ in this interval. (Note: $f'(t)$ is positive on this interval).


Question: Calculate the value of C(7,2) multiplied by C(7,1).


Question: Solve the system of equations $x^2+y^2=\frac{41}{4}$, $y^2+z^2=20$, $z^2+x^2=\frac{89}{4}$ for $x^2, y^2, z^2$. Using these values, calculate the volume $V$ of the tetrahedron.


Question: What is the sum of 147 and 147?


Question: Determine the status (Winning or Losing for the starting player) for positions $n = 0, 1, 2, \dots, 20$. Use the recursive definition: $n=0$ is Losing. For $n>0$, it is Losing if all reachable positions ($n-1$ and $n-4$, if non-negative) are Winning; otherwise it is Winning. Output the list of pairs $(n, \text{status})$.

\begin{aligned} &(0, \text{Losing}), (1, \text{Winning}), (2, \text{Losing}), (3, \text{Winning}), (4, \text{Winning}),

(0, \text{Losing}), (1, \text{Winning}), (2, \text{Losing}), (3, \text{Winning}), (4, \text{Winning}), (5, \text{Losing}


Question: Calculate $75+6$.


Question: Calculate the value of C(7,4) multiplied by C(7,3).


Question: Using the box dimensions $x=4, y=5, z=8$ which define the equifacial tetrahedron, calculate the Volume $V$ and the Total Surface Area $S$ of the tetrahedron. Specifically, determine $V$ and $S$ in simplest form.


Question: In the expression for the maximum real part, $4\sqrt{(75+6)^2 + (9-117)^2}$, we already know $75+6=81$. What is the value of the integer subtraction $9-117$?


Question: Given $x = \left(\frac{1}{2}\right)^3$ and $y = \left(\frac{\sqrt{3}}{2}\right)^3$, calculate the value of $x^2 + y^2$.


Question: What is the number of ways to write 8 as a sum of 3 positive integers? What is the number of ways to write 8 as a sum of 2 positive integers?


Question: Given R=13, r=6, and conditions implying cos A = 7/13 and cos(B-C) = 11/13, calculate the product AB · AC. Use the identity AB·AC = 4R² sin B sin C where sin B sin C = (1/2)[cos(B-C) - cos(B+C)].


Question: Given $x_F - x_E = 184$ and $x_E x_F = 17$ with $0 < x_E < 107$, calculate $x_E$ and then compute $CE = 107 - x_E$.


Question: Calculate the squared distance $OI^2$ between the circumcenter $O$ and incenter $I$ using Euler's theorem $OI^2 = R(R-2r)$ with $R=13$ and $r=6$.


Question: Divide the number 219 by 3 to find the value of n₄.


Question: What is the sum of 195 + 367 + 562 + 900?


Question: Calculate the result of 448 minus 229.


Question: Calculate the limit of the function 480(1+t)/(6t-5) as t approaches 6/5 from below.


Question: Evaluate the numerical value of the expression $4\sqrt{81^2 + (-108)^2}$.


Question: In a set problem with 4 categories, what is the sum of the counts for the Diamond Ring (195), Golf Clubs (367), Garden Spade (562), and Bag of Candy Hearts (900)?


Question: Calculate the integer value of $65^2 + 64^2$, then determine its remainder when divided by 1000.


Question: What is the sum of 2024 and 1?


Question: Calculate the value of the expression 2024 - (2 × 437) - (3 × 234).


Question: Calculate $(1+i)^{13}$.


Question: Calculate the product of 4 and 3 to determine the number of intersections.


Question: Find the least prime $p$ such that the congruence $n^4 \equiv -1 \pmod{p^2}$ has a solution for some positive integer $n$.


Question: Calculate the sum of 195, 367, 562, and 900.


Question: Calculate the number of residents owning exactly 4 items. Use the following values: Total Sum of Item Ownerships = 2024, Total Residents = 900, Exactly 2 Items = 437, Exactly 3 Items = 234. The logical relationship is: Total_Sum - Total_Population = 1*(Count_E2) + 2*(Count_E3) + 3*(Count_E4). Calculate the Count_E4.


Question: Verify if $AP = \frac{25}{13}$ is the correct rational approximation, considering potential simplifications in the $\sqrt{5}$ factor.


Question: Let $B(u, v)$ be a point on the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$. Define $m = v/u$. First, express $u^2$ in terms of $m$. Then compute the value of $BD^2 = 4(u^2 + v^2)$ in terms of $m$.


Question: In a triangle where the inradius is determined by a chain of 2024 circles of radius 1 arranged in the corner, assuming the characteristic dimension is the product N*r, what is the inradius?


Question: Calculate the product $6 \times 13 \times 6$.


Question: Analyze the subtraction game where allowed moves are to remove 1 or 4 tokens from a pile of $n$ tokens. Identify all non-negative integers $n$ that are Losing positions (positions where the current player has no winning strategy). Please list the classification (Winning/Losing) for $n=0, 1, \dots, 15$ and determine the periodicity of these positions modulo $m$.

\text{Losing positions: } n \equiv 0, 2 \pmod 5; \text{ Classification: L, W, L, W, W, L, W, L, W, W, L, W, L, W, W, L;


Question: Calculate the number of residents owning exactly 4 items using the relation $3n_4 = \text{SumSizes} - \text{Population} - n_2 - 2n_3$, where SumSizes=2024, Population=900, $n_2=437$, $n_3=234$.


Question: In a set of 6 distinct items, how many ways can 2 distinct items be chosen?


Question: Calculate $AB \cdot AC$ using the relations: $a = 2R \sin A$, $s = a + r \cot(A/2)$, and $AB \cdot AC = \frac{2rs}{\sin A}$. Inputs: $R=13, r=6, \cos A=7/13$.


Question: Calculate the number of ways Jen's 4 chosen numbers and the Lottery's 4 randomly chosen numbers share exactly 3 common elements. Express this using binomial coefficients $\binom{n}{k} \times \binom{n'}{m'}$ and find the integer value.


Question: In a Cartesian plane, points A(0,16) and D(0,0) lie on a circle. Another horizontal chord GH lies on the line y = -17 with length 184, and the endpoints G and H also lie on this circle. Let E be the projection of H onto the x-axis (so E is the x-coordinate of H). Find the possible values for the x-coordinate of E.


Question: In a system $y = f_n(x)$ and $x = f_m(y)$ where $f_n, f_m$ are waves with $N$ and $M$ periods/humps respectively, the number of intersections is often $N \times M$. Calculate this for the given functions.


Question: How many pairs of non-negative integers \((b, c)\) satisfy \(b+c=200\)?


Question: What is the remainder when 8321 is divided by 1000?


Question: What is the least integer \(M \ge 1\) that has at least 4 distinct prime factors?


Question: Given the number of owners for Diamond Rings (195), Golf Clubs (367), Garden Spades (562), and Candy Hearts (900), what is their sum?


Question: Given a triangle with circumradius $R=13$, inradius $r=6$, and center distances satisfying $IA \perp OI$ where $OI^2=13$, calculate $\cos A$.


Question: Given $s = \frac{39}{4}$, compute $r^2 = \frac{s^2 - 54}{4}$. Express $r^2$ as a simplified fraction $\frac{p}{q}$. Then find the sum $p+q$.


Question: What is the sum of 195, 367, 562, and 900?


Question: In a triangle with circumradius $R=13$ and inradius $r=6$, the distance squared $OI^2$ is given by Euler's theorem as $R(R-2r)$. Calculate $OI^2$.


Question: Calculate 2 multiplied by 234.


Question: In a population of 900 residents where every resident owns at least 1 item, if 437 own exactly 2 items and 234 own exactly 3 items, how many residents own exactly 1 item?


Question: In the Aimeville population of 900, every resident owns between 1 and 4 items. Given that residents owning exactly 1 item = 229, exactly 2 items = 437, and exactly 3 items = 234, what is the number of residents owning exactly 4 items?


Question: Find the least prime number $p$ such that there exists a positive integer $n$ for which $n^{4}+1$ is divisible by $p^{2}$.


Question: Calculate 2024 minus 900 minus 437 minus 468.


Question: What is 219 divided by 3?


Question: Find the least positive integer $m$ such that $m^4 + 1$ is divisible by $289$.


Question: In a triangle with circumradius $R=13$, inradius $r=6$, and condition $IA \perp OI$, calculate the value of $\cos A$.


Question: We need to calculate the total number of distinct ordered triples $(a,b,c)$ formed by the union of the sets $S_a = \{(a,b,c) \mid a=100, b+c=200, a,b,c \in \mathbb{Z}_{\ge 0}\}$, $S_b = \{(a,b,c) \mid b=100, a+c=200, a,b,c \in \mathbb{Z}_{\ge 0}\}$, and $S_c = \{(a,b,c) \mid c=100, a+b=200, a,b,c \in \mathbb{Z}_{\ge 0}\}$. The intersection of any two sets contains exactly one element, $(100,100,100)$. Calculate the size of $S_a \cup S_b \cup S_c$.


Question: Find the least positive integer m such that m^4+1 is divisible by 17^2.


Question: Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$. Compute the exact integer value of the expression $(\alpha^{13}-1)(\beta^{13}-1)$.


Question: Calculate the number of non-negative integer solutions to A + B + C = 8 where 0 <= A, B, C <= 9.


Question: Given AD = 325/22 and DB = 225/22, and the relation DB² = DP · DA, calculate the value of AP. Assume P lies on the segment DA such that D, P, A are collinear and P is between D and A.


Question: Verify if the number of intersections is exactly 24 by accounting for all segments and boundary points, considering the periodicity of $S(x)$ (freq 2) and $C(y)$ (freq 3) combined with the 4-fold branching of $H$. Does the count match $2 \times 3 \times 4$?


Question: How many ways can the integer 8 be expressed as a sum of 3 positive integers?


Question: What is the exact value of the complex number $(1+i)^{13}$?


Question: Given the circle through A(0,16) and D(0,0) has the form $x^2+y^2-16y-190x=0$. If a rectangle EFGH has vertices on this circle such that E and F are on the x-axis with y=-17 (at G and H respectively), and $EF=184$, find the x-coordinate of E, given the order D, E, C, F implies $x_E < x_C=107$.


Question: Let $z = -64 - 64i$. Compute the product $(1-z)(1-\overline{z})$ (where $\overline{z}$ is the complex conjugate of $z$) and provide the remainder when this result is divided by 1000.


Question: What is the number of non-negative integer solutions to a + b + c = 8?


Question: Using the Power of a Point theorem, calculate the length of segment AP. We are given BD = 225/22 and AD = (5*sqrt(793))/22. The point P lies on AD, so DA * DP = DB^2. Find the value of AP = |DA - DP|.


Question: Given the quadratic equation $x_E^2 + 184x_E - 17 = 0$ which arises from the collinearity and cyclic conditions, find the positive root $x_E$, then compute the length $CE = 107 - x_E$.


Question: In a set of 6 items, how many ways are there to choose 2 distinct items?


Question: The number of solutions to $a+b+c=300$ is $\binom{302}{2}$. Calculate $\binom{302}{2}$.


Question: Given $x = -65 - 64i$ and $y = -65 + 64i$, calculate the integer product $x \cdot y$ and report the remainder when this integer is divided by 1000.


Question: In the circle passing through $A(0,16)$ and $D(0,0)$, what is the value of $k$ (x-coordinate of center) if the horizontal chord at height $y=17$ has length 184?


Question: Consider triangle ABC with side lengths AB=5, BC=9, AC=10 inscribed in a circle $\omega$. Let the tangents to $\omega$ at vertices B and C intersect at point D. Calculate the length of the segment BD.


Question: For the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$, let $B(u,v)$ be a point on the curve with slope $s = v/u$ (line $y=sx$). Calculate $OB^2$ in terms of $s$ and find the range of $s^2$ required for a rhombus to exist (both diagonals must intersect the hyperbola).


Question: Calculate the value of the product $(-65-64i)(-65+64i)$ and determine the remainder when divided by 1000.


Question: Calculate the total number of triples by summing permutations of distinct sets $\{k, 100, 200-k\}$ for $k \in \{0, \dots, 99\}$ and the single triple for $k=100$.


Question: Calculate the number of integer solutions to $a_1 + a_2 + a_3 = 8$ given that $0 \le a_1, a_2, a_3 \le 9$. (Hint: This is a standard Stars and Bars problem).


Question: Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a=100$, $a+b+c=300$, and $a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000$.


Question: Let $z = 1+i$. Calculate the value of $z^{13}$.


Question: Given that $a_k, b_k$ are digits ($0 \le a_k, b_k \le 9$) and $S_k = a_k + b_k$, solve the linear Diophantine equation $100 S_1 + 10 S_2 + S_3 = 999$ subject to $0 \le S_k \le 18$ for each $k \in \{1, 2, 3\}$. Return the resulting values of $(S_1, S_2, S_3)$.


Question: Calculate $OI^2$ using the formula $OI^2 = R(R-2r)$ with $R=13$ and $r=6$.


Question: Let $\beta_1 = 1+i$ and $\beta_2 = 1-i$. Calculate the values of $\beta_1^{13}$ and $\beta_2^{13}$ in the form $a+bi$.


Question: In $\triangle OIA$, $\angle OIA = 90^\circ$ (since $IA \perp OI$). Given hypotenuse $OA = R = 13$ and leg $OI^2 = 13$, calculate $AI^2$ using the Pythagorean theorem.


Question: In $\triangle ABD$, we have side $AB=5$, side $BD=\frac{225}{22}$. The angle $\angle DBA$ is equal to $\angle C$ of $\triangle ABC$. Using the Law of Cosines with $\cos C = \frac{13}{15}$, calculate the length of side $AD$.


Question: Given $u = -\frac{7}{24}, v = -\frac{3}{8}, w = -\frac{5}{12}$, calculate the value of the expression $4u + 3v + 2w$.


Question: In how many ways can 8 be expressed as the sum of three non-negative integers $a + b + c = 8$?


Question: Calculate the square of the distance from the origin $(0,0)$ to the point $C\left(\tfrac{1}{8}, \tfrac{3\sqrt{3}}{8}\right)$, express the result as an irreducible fraction $\tfrac{p}{q}$ where $p$ and $q$ are positive integers, and return the sum $p+q$.


Question: Given $x,y,z > 0$ satisfy $xy+yz+zx=27$ and $xyz=23$. Let $s = x+y+z$. What is the maximum possible value of $s$?


Question: Find the least prime $p$ such that the congruence $n^4 \equiv -1 \pmod{p^2}$ has a solution in integers $n$.


Question: Calculate the sum of the squares of the items in the list {5, 7, 9, 9}.


Question:
Solve the following system of linear equations for $r_{in}$ where $K$ is a non-zero constant:
1) $(r_{in} - 34) K = 476$
2) $(r_{in} - 1) K = 4046$
Express the resulting value of $r_{in}$ as an irreducible fraction $m/n$, where $m$ and $n$ are coprime integers. Then calculate the sum $m+n$. Provide the final integer answer.


Question: Verify if $657/64$ is the maximal possible value for $x^2+y^2+z^2$ given $xy+yz+zx=27$ and $xyz=23$, specifically confirming the case where two sides are equal yields the maximum.


Question: What is the sum of 195, 367, 562, and 900?


Question: Calculate the value of 2 multiplied by 437 plus 3 multiplied by 234.


Question: Calculate the value of the expression (448 - 229) divided by 3. This represents the number of residents who own all four items.


Question: Subtract 1576 from the total item sum 2024 to find the value of $N_1 + 4N_4$.


Question: Calculate the sum of the reciprocals of 200, 240, and 300. Specifically, compute 1/200 + 1/240 + 1/300.


Question: In the family of unit segments described above, let the segment be \(\overline{AB}\) with endpoints \(A=(\tfrac{1}{2},0)\) and \(B=(0,\tfrac{\sqrt{3}}{2})\). Find the parameter \(\theta \in (0, \pi/2)\) such that the segment's endpoints match \(A\) and \(B\). Then compute the envelope coordinates \(X = \cos^3 \theta\) and \(Y = \sin^3 \theta\) for this \(\theta\). Provide the resulting pair \((X,Y)\).


Question: Calculate the roots of the quadratic equation $z^2 - 2z + 2 = 0$.


Question: Calculate the values of $|75+117i|$ and $|96+144i|$.


Question:
Let $u, v, w$ be real numbers satisfying the following system of linear equations:
$$u - v - w = \frac{1}{2}$$
$$-u + v - w = \frac{1}{3}$$
$$-u - v + w = \frac{1}{4}$$
Solve for the exact values of $u, v,$ and $w$.


Question: Calculate the sum of squares of 5, 7, 9, and 9.


Question: Calculate the value of $65^2 + 64^2$ and report the remainder when this value is divided by 1000.


Question: How many ways can one choose 2 distinct items from a set of 6 distinct items?


Question: In a regular dodecagon with 12 vertices, how many pairs of vertices are connected by a line passing through the center of the dodecagon (diametral pairs)?


Question: Calculate the sum $1/200 + 1/240 + 1/300$.


Question: In a linear combination, calculate $4 \times \left(-\frac{7}{24}\right) + 3 \times \left(-\frac{3}{8}\right) + 2 \times \left(-\frac{5}{12}\right)$.


Question: Solve the linear system $A - B - C = 1/2$, $-A + B - C = 1/3$, $-A - B + C = 1/4$. Output the values of $A$, $B$, and $C$.


Question: For a fixed positive integer k, how many finite nonempty sets B of positive integers satisfy the condition that the maximum element of B is exactly k?


Question: Let $H(u) = 4 g(f(u))$ for $u \in [0, 1]$, where $f(x) = ||x| - 1/2|$ and $g(x) = ||x| - 1/4|$. Compute the simplified expression for $H(u)$, determine its periodicity, range, and identify the coordinates of its local maxima and minima in $[0, 1]$.


Question: Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $289$.


Question: On the hyperbola defined by $\frac{x^2}{20} - \frac{y^2}{24} = 1$, let $A$ and $B$ be two points such that $OA \perp OB$ (where $O$ is the origin). Let $D$ be the reflection of $B$ across the origin. Calculate the infimum of the value $BD^2$.


Question: Consider the system of equations: $2x^2 + 2xz = 27$ and $x^2 z = 23$ with $x, z > 0$. Eliminate $z$ to form a cubic equation in $x$. Find the positive real roots of this cubic equation and the corresponding value of $z$ for each root.


Question: Calculate the sum 1/200 + 1/240 + 1/300.


Question: What is the sum of the elements of the set $\{11, 10, 9, 8, 7, 6, 4\}$?


Question: Calculate the sum of the following integers: 11 + 10 + 9 + 8 + 7 + 6 + 4.


Question: Calculate the sum of the squares of the numbers 5, 7, 9, and 9.


Question: Calculate $AB \cdot AC$ given $R=13, r=6, \sin^2(A/2) = 3/13$.


Question: Calculate the product $2 \times 3 \times 5 \times 7$. Return this product.


Question: What is the sum of the squares of 5, 7, 9, and 9?


Question: Calculate the sum of 25 and 8.


Question: In a relationship where 1/s = 1/a + 1/b + 1/c, with a=200, b=240, and c=300, what is the value of s?


Question: In the equation $\sum_{k \in A} 2^{k-1} = 2024$, where $A$ is a set of distinct positive integers, what is the value of $\sum_{k \in A} k$?


Question:
Solve the cubic equation $x^3 - 27x + 46 = 0$ for positive real roots.
For each positive root $x$, calculate the squared radius $r^2 = \frac{1}{4}(2x^2 + (23/x^2)^2)$ corresponding to the box dimensions $x, x, z$ (where $z=23/x^2$).
Report the maximum value of $r^2$ as an irreducible fraction $p/q$, and the integer value $p+q$.


Question: In a set of 6 distinct main diagonals of a regular dodecagon, how many ways are there to choose 2 diagonals?


Question: Simplify log2(x/(yz)) = 1/2 into an equation involving log2(x), log2(y), and log2(z).


Question: Calculate the value of the expression $4A + 3B + 2C$ using the values $A = -\frac{7}{24}$, $B = -\frac{3}{8}$, and $C = -\frac{5}{12}$. Provide the result as a fraction.


Question: Express the integer 2024 as a sum of distinct powers of 2 (i.e., its binary expansion). Provide the list of exponents $b$ such that $2024 = \sum 2^b$.


Question: Express 2024 as a sum of distinct powers of 2.


Question: Find the least prime number p such that there exists a positive integer n where n^4 + 1 is divisible by p^2.


Question: Given the equations $\log_2(x/yz) = 1/2$, $\log_2(y/xz) = 1/3$, and $\log_2(z/xy) = 1/4$. Let $a = \log_2 x$, $b = \log_2 y$, and $c = \log_2 z$. Rewrite each equation using the properties of logarithms to form a linear system in terms of $a, b,$ and $c$. List the three resulting linear equations.


Question: Compute the value of the sum $S = \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$.


Question: In the system of equations, adding the first two equations yields $-2c = \frac{1}{2} + \frac{1}{3}$. Calculate the value of $c$.


Question: Calculate the sum of squares for the list 3, 9, 9, 9.


Question: In the list {5, 7, 9, 9}, square each number (5²=25, 7²=49, 9²=81, 9²=81) and sum all squared values to get the total sum of squares.


Question: In a coordinate system where $D$ is at $(0,0)$ and the line containing $D,E,C,F$ is the x-axis. Let $A$ be at $(0,16)$ and vertices $H(x_E, 17)$, $G(x_F, 17)$ of a rectangle with side length $EF=184$ be such that $A, D, H, G$ lie on a circle. Find the value of the product $x_E \cdot x_F$.


Question:
Solve the following system of linear equations for the values of $A, B,$ and $C$:
1) $A - B - C = \frac{1}{2}$
2) $B - A - C = \frac{1}{3}$
3) $C - A - B = \frac{1}{4}$
Return the values of $A, B,$ and $C$.


Question: Find a list of 4 positive integers $\{x_1, x_2, x_3, x_4\}$ (sorted) such that $x_1+x_2+x_3+x_4 = 30$, the frequency of 9 is greater than any other element's frequency, the median $(x_2+x_3)/2$ is an integer not present in the list.


Question:
Solve the following system of linear equations for variables $a, b, c$:
$a - b - c = \frac{1}{2}$
$-a + b - c = \frac{1}{3}$
$-a - b + c = \frac{1}{4}$


Question: Calculate the sum of the elements: $11 + 10 + 9 + 8 + 7 + 6 + 4$.


Question: Calculate the value of s given that 1/s = 1/200 + 1/240 + 1/300.


Question: Evaluate the expression $4a + 3b + 2c$ using the values $a = -\frac{7}{24}, b = -\frac{3}{8}, c = -\frac{5}{12}$. Express the result as a positive irreducible fraction $\frac{m}{n}$, then calculate $m+n$ and provide the answer.


Question: Given $c = -5/12$, $a - b = \frac{1}{12}$, and $a + b = -\frac{2}{3}$, calculate the value of $|4a + 3b + 2c|$.


Question: Calculate the value of 107 minus the smaller positive root of the equation t^2 + 184t - 17 = 0.


Question: Evaluate $65^2 + 64^2$, then compute the remainder when that sum is divided by 1000.


Question: In a list of positive integers with sum 30, unique mode 9, and median an integer not in the list, identify the list.


Question: In triangle ABC with sides AB=5, BC=9, and AC=10, let D be the intersection of the tangents to the circumcircle at B and C. Calculate the length of the segment AD.


Question: Calculate the sum of squares $65^2 + 64^2$ and give the result modulo 1000.


Question: Calculate the value of s given by the formula $s = \frac{200 \cdot 240 \cdot 300}{(200 \cdot 240) + (240 \cdot 300) + (300 \cdot 200)}$.


Question: What is the exact value of $(1+i)^{13}$?


Question: Express 2024 as a sum of distinct powers of 2. For example, '2024 = 1024 + 512 + ...'.


Question: How many ways are there to choose three digits $d_1, d_2, d_3$ (where each digit is between 0 and 9 inclusive) such that $d_1 + d_2 + d_3 = 8$?


Question: Calculate the value of $(1+i)^{13}$.


Question: Find the least positive integer $m$ such that $m^4 + 1$ is divisible by $289$.


Question: Calculate the value of the expression $F(u) = \frac{1920(1+u)}{24-20u}$ when $u$ is equal to $5/6$.


Question: Calculate the sum 4 + 6 + 7 + 8 + 9 + 10 + 11.


Question: In a regular dodecagon with 12 vertices, how many pairs of diametrically opposite vertices exist?


Question: What is the sum of the integers 4, 6, 7, 8, 9, 10, and 11?


Question: In a 5x5 grid, let the number of valid maximal configurations be given by the formula $1 + (\sum_{k=1}^5 \binom{5}{k})^2$. Evaluate this expression.


Question: Calculate the sum of the distinct positive integers $\{4, 6, 7, 8, 9, 10, 11\}$.


Question: Calculate the value of $4 \times \left(-\frac{7}{24}\right) + 3 \times \left(-\frac{3}{8}\right) + 2 \times \left(-\frac{5}{12}\right)$. Provide the result as a simplified rational number.


Question: Express 2024 as a sum of distinct powers of 2. Return the set of exponents $\{k\}$ such that $2^{k}$ is included in the sum for $2024$.


Question: Calculate the expression $1 + \left(\sum_{k=1}^4 \binom{5}{k}\right)^2 + 1$. Recall $\binom{5}{0} = 1$ and $\binom{5}{5} = 1$.


Question: In a regular 12-sided polygon, how many pairs of vertices are diametrically opposite?


Question: How many distinct pairs of diameters can be formed from a set of 6 diameters?


Question:
Solve the linear system:
1) $A - B - C = 1/2$
2) $-A + B - C = 1/3$
3) $-A - B + C = 1/4$
Find the values of $A, B, C$.


Question: In a regular dodecagon, there are 6 distinct diameters connecting opposite vertices. How many rectangles can be formed by selecting 2 distinct diameters?


Question: What is the least integer b >= 2 such that omega(b-1) >= 4?


Question: In a triangle with sides $a=9$, $b=10$, and $c=5$, what is the value of $\cos A$? (Use the formula $\cos A = \frac{b^2+c^2-a^2}{2bc}$)


Question: In a regular dodecagon, there are 6 diameters connecting opposite vertices. If these diameters are ordered by angle (with adjacent ones separated by 30°), how many pairs of these 6 diameters are perpendicular to each other?


Question: Write 2024 as a sum of distinct powers of 2: $2^{e_1} + \dots + 2^{e_k} = 2024$. Return the set of exponents $\{e_1, \dots, e_k\}$.


Question: Using the Stars and Bars theorem, calculate the number of non-negative integer solutions to the equation $a+b+c=8$.


Question: Find the list of positive integers satisfying: sum of elements is 30, the unique mode is 9, and the median is a positive integer that does not appear in the list.


Question:
Solve the following system of congruences for non-negative integers $d_3, d_2, d_1, d_0$ representing digits (values $0-9$, $d_3 \neq 0$):
1) $2d_2 + 3d_1 + d_0 \equiv 1 \pmod 7$
2) $6d_3 + 3d_1 + d_0 \equiv 5 \pmod 7$
3) $6d_3 + 2d_2 + d_0 \equiv 4 \pmod 7$
4) $6d_3 + 2d_2 + 3d_1 \equiv 6 \pmod 7$
Find the tuple $(d_3, d_2, d_1, d_0)$ that maximizes the 4-digit number $N$ formed by these digits. Output the number N.


Question: Calculate $30 \times 30$ and add 2 to the result.


Question: Consider the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$. If a line through the origin with slope $m$ intersects the hyperbola at a point $P(x,y)$, express $x^2+y^2$ in terms of $m$. Specifically, derive the function $f(u) = \frac{1920(1+u)}{24-20u}$ where $u=m^2$ represents the squared distance contribution, assuming perpendicular diagonals define a valid rhombus. Confirm the range of $u$ allowed for both vertices to exist on the hyperbola.


Question: Find the distinct powers of 2 that sum to 2024. Return the set of exponents $k$ for each $2^k$ in the sum.


Question:
Solve the following system of linear equations for $u, v,$ and $w$:
\begin{align*}
u - v - w &= \frac{1}{2} \\
-u + v - w &= \frac{1}{3} \\
-u - v + w &= \frac{1}{4}
\end{align*}


Question: Verify the result $bc=468$ is consistent with all triangle properties given $R=13, r=6$ and $IA \perp OI$. Specifically, confirm the calculated $ab \cdot ac$.


Question: Evaluate the expression: (12*\sqrt{2146} + 12*\sqrt{13}) * \cos\left(\frac{\arctan\left(\frac{39}{25}\right) + \arctan\left(\frac{3}{2}\right)}{2}\right).


Question: Write the decimal number 2024 as a sum of distinct powers of 2. List the exponents $e$ (where the terms are $2^e$) that appear in this sum.


Question: Calculate the number of non-negative integer solutions to the equation $t_1 + t_2 + t_3 = 8$.


Question: In a triangle $ABC$, a chain of $N$ circles of equal radius $r$ is packed such that the first circle is tangent to side $AB$, the last circle is tangent to side $BC$, and the entire chain is tangent to side $AC$. Derive the equation that relates $N$, $r$, and the inradius $r_{in}$ of triangle $ABC$.

(r_{in} - r) \left( \cot \frac{A}{2} + \cot \frac{C}{2} \right) = 2r(N-1)


Question: In triangle ABC, side BC = 9 and cos(angle A) = 11/25. Point D is the intersection of tangents to the circumcircle at B and C. In triangle DBC, the base angles at B and C are both equal to angle A. Calculate the length of segment DB.


Question: Compute the value of $4(-\frac{7}{24}) + 3(-\frac{3}{8}) + 2(-\frac{5}{12})$. Express the result as a simplified fraction $\frac{m}{n}$. Also determine if $m$ and $n$ are relatively prime positive integers.


Question: Solve for the integer values of the sums $S_1 = a+d$, $S_2 = b+e$, and $S_3 = c+f$ subject to the constraints $a,b,c,d,e,f \in \{0,1,\dots,9\}$ and the equation $100S_1 + 10S_2 + S_3 = 999$.


Question: Using the Power of a Point theorem for point $D$ with respect to circle $\omega$, where secant $DPA$ satisfies $DB^2 = DP \cdot AD$, calculate the length of $AP$ given $DB = \frac{225}{22}$ and $AD = \frac{325}{22}$. Express the result as a reduced fraction $\frac{m}{n}$.


Question: Verify that the number of solutions to $k(k-1) \equiv 0 \pmod M$ in $1 \le k \le M$ is exactly $2^{\omega(M)}$, and confirm $k=1$ is always excluded by the range $k \ge \lceil \sqrt{b} \rceil$. Then conclude.


Question: In a list of positive integers with sum 30, unique mode 9, and median being a positive integer not present in the list, why must the number of elements $n$ be even?

\text{If } n \text{ is odd, the median is an element of the list.}


Question: In a $2\times3$ grid with digit entries $t_1,t_2,t_3$ (row 1) and $b_1,b_2,b_3$ (row 2), suppose the sum of the two 3-digit numbers formed by the rows equals 999. That is, $(100t_1+10t_2+t_3) + (100b_1+10b_2+b_3) = 999$. Given that all entries are digits $0 \dots 9$, what are the values of the column sums $s_j = t_j + b_j$ for $j=1,2,3$?


Question: Given points A=(1/2, 0) and B=(0, √3/2), find the slope m and y-intercept b of the line passing through A and B. Provide the equation of the line in the form y = mx + b.


Question: Divide the integer 5694 by 1000 to find the quotient Q and the remainder R. Then calculate the value of Q + R. Report the result.


Question: Calculate OC² = x² + y² for C(1/8, 3√3/8). Express OC² as p/q where p, q are coprime. Compute p+q.


Question: Given a triangle with circumradius $R=13$ and inradius $r=6$, what is the value of $OI^2$ (the square of the distance between the circumcenter and incenter)?


Question: Calculate the value of the binomial coefficient \(\binom{10}{2}\).


Question:
Construct the greatest four-digit integer $N$ formed by digits $d_1, d_2, d_3, d_4$ such that $d_i \in \{0, \dots, 9\}$, $d_1 \neq 0$, and the following congruences hold modulo $7$:
$$d_1 \equiv 5, \quad d_2 \equiv 6, \quad d_3 \equiv 2, \quad d_4 \equiv 4.$$
Return $N$.


Question: Solve the system of equations: x^(2/3) + y^(2/3) = 1 and √3x + y = √3/2 for x, y > 0. Find the value of x² + y².


Question: In triangle $ABC$ with $a=9, b=10, c=5$, let $D$ be the intersection of tangents to circumcircle at $B$ and $C$. The line $AD$ intersects the circumcircle at $P$. Calculate the length $AP$.


Question: Based on the configuration of "sequentially tangent circles tangent to two sides of a triangle", what is the general formula relating the number of circles $N$, their common radius $R$, and the inradius $r_{in}$? Specifically, does $N \cdot R$ relate to $r_{in}$ linearly or affinely?


Question: Let $C_1, C_2, C_3$ be integers such that $0 \le C_i \le 18$ for all $i$. Solve the equation $100 C_1 + 10 C_2 + C_3 = 999$. Return the ordered triple $(C_1, C_2, C_3)$.


Question: Determine the dimensions of the rectangular box with surface area 54 and volume 23 such that the dimensions are rational numbers.


Question: Consider a four-digit positive integer $N$ with digits $A, B, C, D$ (so $N = 1000A + 100B + 10C + D$). The problem states that whenever one of these digits is changed to $1$, the resulting number is divisible by $7$. Write down the four specific linear congruences modulo $7$ that represent these conditions involving $A, B, C, D$.

\begin{aligned}
&2B + 3C + D \equiv 1 \pmod 7 \\
&6A + 3C + D \equiv 5 \pmod 7 \\
&6A + 2B + D \equiv 4 \pmod 7 \\
&6A +

\begin{aligned}
6 + 2B + 3C + D &\equiv 0 \pmod 7 \\
6A + 2 + 3C + D &\equiv 0 \pmod 7 \\
6A + 2B + 3 + D &\equiv 0 \pmo


Question: Find the greatest four-digit positive integer N such that changing any one of its digits to 1 results in a number divisible by 7. Report the value of N.


Question: What is the sum of 5 and 694?


Question: In a triangle with $R=13$ and $r=6$, if $IA \perp OI$, find $\sin^2(A/2)$ and the side length $a$. Use $IA = r/\sin(A/2)$ and $a = 2R\sin A$.


Question: Let $N$ be a four-digit integer such that changing any of its digits to $1$ produces a number divisible by $7$. Find the value of $N \pmod 7$.


Question: Analyze the constraints (Sum=30, Mode=9, Median integer not in list) to find the list size $n$. Check if $n=4$ is the only possible even size, and if so, identify the unique list of numbers satisfying all conditions. Finally, calculate the sum of the squares of these numbers.


Question: Solve the linear system: A - B - C = 1/2, -A + B - C = 1/3, -A - B + C = 1/4 for the variables A, B, and C.


Question: Solve the linear system for A, B, C where the coefficients are derived from the expansion of log terms as above.


Question: In triangle ABD, we have side $AB=5$, side $DB=\frac{225}{22}$, and $\angle ABD = A + B$ where $\cos(A+B) = -\frac{13}{15}$. Find the length of side $AD$ using the Law of Cosines ($DB^2 = AB^2 + AD^2 - 2 \cdot AB \cdot AD \cdot \cos(A+B)$).


Question: Calculate $bc$ given that $x = s-a$ is the valid root of the equation $\sqrt{30}x^2 - 78x + 36\sqrt{30} = 0$ that satisfies the triangle existence condition, and $bc = x^2 + 348$.


Question: For $a=5$, find the largest digits $b, c, d \in \{0..9\}$ satisfying $b \equiv 6 \pmod 7$, $c \equiv 2 \pmod 7$, and $d \equiv 4 \pmod 7$.


Question: Based on the configuration of a chain of $N$ circles of radius $R$ in a triangle corner, what is the value of the inradius $r_{in}$ expressed as $\frac{m}{n}$ using the second case where $N=2024$ and $R=1$?


Question: Evaluate the numerical value of the expression: $4 \times \left(-\frac{7}{24}\right) + 3 \times \left(-\frac{3}{8}\right) + 2 \times \left(-\frac{5}{12}\right)$. Simplify the result to an irreducible fraction $\frac{m}{n}$ (where $m, n$ are positive integers and $\gcd(m,n)=1$), compute its absolute value, and finally determine the integer sum $m+n$.


Question: Given $C=(\frac{1}{8}, \frac{3\sqrt{3}}{8})$ and $O=(0,0)$, calculate $OC^2$. Simplify the result to a fraction $\frac{p}{q}$ in lowest terms where $p,q$ are positive integers, and return the value of $p+q$.


Question: In the fractions $33/4$ and $33/7$, calculate the absolute difference. Express this difference as a simplified fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Return the value of $m+n$.


Question: In the division of 5694 by 1000, what is the sum of the quotient and the remainder?


Question: Find the point of tangency between the line $\sqrt{3}x + y = \sqrt{3}/2$ and the astroid $x^{2/3} + y^{2/3} = 1$.


Question: Calculate the number of $b$-eautiful integers for \(b=1156\).


Question: For a four-digit integer $N$ with the property that changing any single digit to $1$ results in a number divisible by $7$, what is the value of $N \pmod 7$?


Question: Given the constraints $xy+yz+zx=27$ and $xyz=23$ for positive real numbers $x,y,z$, calculate the maximum possible value of the sum of squares $S = x^2+y^2+z^2$.


Question: Calculate the number of valid "b-beautiful" integers for $b=211$. Specifically, find the number of integers $k \in \{1, \dots, 210\}$ such that $k(k-1)$ is divisible by 210 and $k^2 \ge 211$.


Question: Determine the digits d3, d2, d1, d0 of the greatest four-digit integer N satisfying the property that replacing any digit with 1 results in a number divisible by 7. Provide the digits in order d3, d2, d1, d0.


Question: In a torus of major radius 6 and minor radius 3 resting on a sphere of radius 11, what are the two radii of the circles of contact?


Question: Determine the possible values for the frequency $k$ of the number 9 in the list, given the sum is 30, items are positive integers, and 9 is the unique mode.


Question: For $b=211$, determine the number of $b$-beautiful integers. Note that this is equivalent to counting integers $s$ in the range $[\lceil \sqrt{211} \rceil, 210]$ such that $s(s-1)$ is divisible by $210$.


Question: What is the sum of the radii of the circle arrangements?


Question: Under the constraints $xy+yz+zx=27$ and $xyz=23$, determine the dimensions $(x,y,z)$ that maximize the value $x^2+y^2+z^2$.


Question: What is the sum of 21 and 49?


Question: Given the line \(y = -\sqrt{3}x + \frac{\sqrt{3}}{2}\) and the curve \(x^{2/3} + y^{2/3} = 1\), find the coordinates of their intersection point \(C\).


Question: Using the simplification $\frac{9b+7}{b+7} = 9 - \frac{56}{b+7}$, identify all integer values of $b$ greater than 9 such that $b+7$ divides 56.


Question: Solve the system of modular equations $6 d_3 + 2 d_2 + 3 d_1 + 1 \equiv 0$, $6 d_3 + 2 d_2 + 3 + d_0 \equiv 0$, $6 d_3 + 2 + 3 d_1 + d_0 \equiv 0$, $6 + 2 d_2 + 3 d_1 + d_0 \equiv 0 \pmod 7$ to find the unique residues of $d_3, d_2, d_1, d_0 \in \{0, \dots, 9\}$ satisfying $d_3 \neq 0$ and maximizing $N$.


Question: Given an integer base $b > 9$, the number written as $17_b$ represents digits 1 and 7 in that base, and $97_b$ represents digits 9 and 7. Using standard positional notation rules ($d_k \dots d_0 = \sum d_i b^i$), calculate the integer value of $17_b$ and $97_b$ as polynomials in $b$.


Question: In a trapezoid with area 72 and height 6, find the sum of the lengths of the parallel sides (r + s) using the formula Area = 1/2 * (r + s) * height.


Question: The radius of the inscribed circle of the trapezoid is 3. What is the height of the trapezoid?


Question: Calculate the value of $r_{in}$ given $h_b = \frac{45696}{11}$ and $\sin(B/2) = \frac{11}{1179}$. Simplify the result to the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Report the value of $m+n$.


Question: Identify all integer triplets (c, v, s) such that c + v + s = 9, c, v, s are integers greater than or equal to 1, and c > v > s.


Question: In an isosceles trapezoid with an inscribed circle of radius 3, what is the height of the trapezoid?


Question:
The equation $12x^2 - xy - 6y^2 = 0$ factors as $(4x - 3y)(3x + 2y) = 0$.
Branch 1: $4x - 3y = 0 \implies x = 3k, y = 4k$. Calculate the number of integers $k$ such that $-100 \le 3k \le 100$ and $-100 \le 4k \le 100$.
Branch 2: $3x + 2y = 0 \implies x = 2m, y = -3m$. Calculate the number of integers $m$ such that $-100 \le 2m \le 100$ and $-100 \le -3m \le 100$.
Return the count for Branch 1 and the count for Branch 2.


Question: Consider the equation $|25+20i-z|=5$. Identify the complex number representing the center of the circle and the real number representing the radius.

\text{Center: } 25+20i, \text{ Radius: } 5


Question: Find the sum of all integers $b$ such that $b > 9$ and $(b+7)$ divides 56.


Question: In an isosceles trapezoid with parallel sides $r$ and $s$, height $h=6$, area $A=72$, and an inscribed circle (implying $r+s = 2l$ where $l$ is the leg length). Calculate the value of $r^{2}+s^{2}$. Include the derivation using $l^{2} = h^{2} + (\frac{r-s}{2})^{2}$.


Question: How many integer values of $k$ satisfy the condition that the pair $(3k, 4k)$ has both coordinates in the range $[-100, 100]$? Specifically, find the number of integers $k$ such that $-100 \le 3k \le 100$ and $-100 \le 4k \le 100$. Return the count as a single integer.


Question: A positive integer n is b-beautiful if n has exactly two digits in base b (digits d1, d0 with 1<=d1<b, 0<=d0<b) and sum of digits equals sqrt(n). This implies n = (d1+d0)^2. Combining n = d1*b + d0, we get b = ((d1+d0)^2 - d0)/d1. For a given b, a pair (d1, d0) is valid if 1<=d1<b, 1<=d0<b, and the above equality holds. Please compute the count of such valid pairs (d1, d0) for each integer b from 5 to 30 inclusive. Return the list of (b, count).


Question: Find the number of integer values of m such that the coordinates $(-2m, 3m)$ satisfy the constraints $|x| \le 100$ and $|y| \le 100$.


Question: Given b from 5 to 50, calculate the number of "b-beautiful" integers for each. Recall a number n is b-beautiful if n = d1*b + d0 (digits d1,d0) and d1+d0 = sqrt(n). Return a list of pairs (b, count).


Question: Based on the definition that if $a$ divides $b$, then $a$ divides any linear combination $cb+d$, show that for $17_b$ to divide $97_b$, $b+7$ must divide 56. (Hint: Use the fact that if $(b+7) \mid (97_b)$ and $(b+7) \mid (9 \times 17_b)$, then $(b+7) \mid (97_b - 9 \times 17_b)$).


Question: What is the sum of 124 and 386?


Question: The radius of the torus tube is $r=3$ and the major radius is $R=6$. When a torus rests on a sphere of radius 11, it is tangent along two distinct circles on the sphere. Assuming the difference between these radii is equal to the diameter of the torus tube, calculate the sum of the numerator and denominator of the difference expressed as a fraction in simplest form.


Question: Calculate the sum $S = \frac{9!}{6!2!1!} + \frac{9!}{5!3!1!} + \frac{9!}{4!3!2!}$ and return the value of $S \pmod{1000}$.


Question: Calculate the integer value of the expression: $9 \times (-7)^2 + 7 \times (-7) + 1$.


Question: Let $A$ be the integer value of the base-$b$ representation $17_b$ and $B$ be the integer value of $97_b$. Calculate the integer value of $B - 9A$.


Question: Find the sum of all integer values $b > 9$ such that $(b+7)$ is a divisor of 56.


Question: Calculate 2304 - 2025


Question: Given two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$ such that the vectors $\vec{OA}$ and $\vec{OB}$ are orthogonal ($x_1 x_2 + y_1 y_2 = 0$), find the minimum value of $4(x_2^2 + y_2^2)$ considering the constraint that finite points $A$ must exist.


Question: Given the circle passes through D(-107, 0), A(-107, 16), H(-x, -17), G(184-x, -17), calculate the value of x (CE).


Question: Calculate the sum of 21 and 49.


Question: Evaluate the expression $(9b + 7) - 9(b + 7)$.


Question: Find all positive integer divisors of 393.


Question: Using the fact that the triangle $ABC$ accommodates 8 circles of radius 34 and 2024 circles of radius 1 in a "sequentially tangent" manner in the "same manner," calculate the inradius $\rho$. Formulate the algebraic relationship implied by this "same manner" constraint (where linear or geometric scaling holds) and solve for $\rho$ in simplest fractional form $m/n$.


Question: Compute the value of the multinomial coefficient $\frac{9!}{6! \cdot 2! \cdot 1!}$.


Question: List all positive integers $d$ that divide 56 evenly and satisfy $d > 16$.


Question: Calculate the multinomial coefficient $\frac{9!}{5! \cdot 3! \cdot 1!}$.


Question: Find all ordered triples (C, V, S) of positive integers such that C + V + S = 9 and C > V > S. List them explicitly.


Question: Calculate A = 9!/(6!2!1!), B = 9!/(5!3!1!), C = 9!/(4!3!2!). Compute Sum = A + B + C. Return (Sum % 1000).


Question: Calculate the value of the multinomial coefficient \frac{9!}{4!3!2!}.


Question: Using the area of 72 and height of 6, calculate the sum of the parallel side lengths $r+s$ (since $\text{Area} = \frac{1}{2}(r+s)h$).


Question: Given an isosceles trapezoid with parallel sides of lengths $r$ and $s$ such that $r+s=24$, a height of 6, and a leg length of 12. Using the relationship $\text{Leg}^2 = \left(\frac{|r-s|}{2}\right)^2 + \text{Height}^2$, calculate the value of $(r-s)^2$.


Question: An isosceles trapezoid has an inscribed circle of radius 3. What is the height of the trapezoid?


Question: Given positive integers c, v, s such that c + v + s = 9 and c > v > s, find all possible integer triples (c, v, s). List them explicitly.


Question: Calculate (12 - 6√3)² + (12 + 6√3)².


Question: In the configuration where N circles of radius r fit in a corner of angle B to touch the opposite side, with csc(B/2) = 1179/11, what is the inradius of the triangle if the configuration is symmetric (isosceles triangle)?


Question: In an isosceles trapezoid with an inscribed circle of radius 3, the height is 6. The area is 72. Calculate the sum of the lengths of the parallel sides.


Question: In base $b$, the number $17_b$ has digits 1 and 7. Calculate its value as a polynomial in $b$. Similarly, the number $97_b$ has digits 9 and 7. Calculate its value as a polynomial in $b$.


Question: Calculate the value of $m+n$ given $h_b = 45696/11$, $\sin(B/2) = 11/1179$, and $r_{in} = h_b \frac{\cos^2(B/2)}{\sin(B/2) + \cos^2(B/2)}$ where $\cos^2(B/2) = 1 - \sin^2(B/2)$.


Question: In this expression: $h_b = \frac{45696}{11}$ and $\sin(B/2) = \frac{11}{1179}$, compute the value of $r_{in} = \frac{45696}{11} \cdot \frac{\cos^2(B/2)}{\sin(B/2) + \cos^2(B/2)}$.


Question: Calculate the multinomial coefficient for the distribution of 9 items into groups of size 6, 2, and 1. Use the formula 9! / (6! * 2! * 1!). Return the calculated integer.


Question: Determine the mathematical relationship between the number of circles $N$, radius $r$, and the inradius $r_{in}$ of Triangle ABC for this specific chain configuration. Use the given pairs $(N=8, r=34)$ and $(N=2024, r=1)$ to solve for $r_{in} = m/n$.


Question: In a configuration where a torus (minor radius 3, distance to axis 6) is externally tangent to a sphere (radius 11) along a circle, what are the radii of the two possible circles of tangency?


Question: In an isosceles triangle $ABC$ with altitude $h_b = 45696/11$ and angle parameter $\sin(B/2) = 11/1179$, compute the inradius $r_{in}$ using the formula $r_{in} = h_b \frac{1-\sin^2(B/2)}{\sin(B/2)+1-\sin^2(B/2)}$. Let $r_{in} = m/n$ where $m,n$ are relatively prime positive integers. Find the value of $m+n$.


Question: Simplify the equation $(8k+73)^2 = 25(8k^2+56k+100)$ into the standard quadratic form $Ak^2 + Bk + C = 0$. Return the integer coefficients $A, B, C$ for the left-hand side minus the right-hand side equals zero.


Question: Find all integer triples $(c, v, s)$ such that $c + v + s = 9$ and $c > v > s \ge 1$.


Question: Calculate the values of the multinomial coefficients $\frac{9!}{6!2!1!}$, $\frac{9!}{5!3!1!}$, and $\frac{9!}{4!3!2!}$. Then compute their sum $S$. Also provide the remainder when $S$ is divided by 1000.


Question: Compute the multinomial coefficient M2 = 9! / (5! * 3! * 1!). Show the full arithmetic steps and provide the final integer result in a box.


Question: Calculate the value of 9! divided by (6! * 2! * 1!). Provide the integer result.


Question: Based on the standard geometric model for arranging N equal circles of radius r sequentially tangent in a triangle corner (from vertex B to side AC), derive the value of the inradius r_in such that both configurations (N=8,r=34 and N=2024,r=1) fit exactly.


Question: Calculate the sum of 128 and 693.


Question: Calculate the value of the multinomial coefficient 9!/(5! * 3! * 1!).


Question: In an isosceles trapezoid with an inscribed circle of radius 3 and an area of 72, what is the sum of the lengths of the two parallel sides?


Question: Given the condition that $b + 7$ must divide 56 and $b > 9$, calculate the sum of all integer values of $b$ that satisfy this requirement.


Question:
Given an isosceles trapezoid with parallel sides $r$ and $s$ such that $r+s=24$. Due to the inscribed circle property, the sum of the legs equals the sum of the bases, so each leg has length 12. The height of the trapezoid is 6.
Using the Pythagorean theorem relationship $12^2 = 6^2 + \left(\frac{|r-s|}{2}\right)^2$, calculate $(r-s)^2$.
Then use the identity $(r-s)^2 + (r+s)^2 = 2(r^2 + s^2)$ to find the exact value of $r^2 + s^2$.


Question: In how many ways can we assign flavors to 9 players with the distribution (4, 3, 2) for (Chocolate, Vanilla, Strawberry)?


Question: Given $\sin(B/2) = 11/1179$, compute the value of $\cos^2(B/2) = 1 - \sin^2(B/2)$ and express it as a simplified fraction $A/B$. Also compute the values of the denominator term $D = \sin(B/2) + \cos^2(B/2)$ and the numerator term $N = h_b \cdot \cos^2(B/2)$ using $h_b = 45696/11$. Finally, calculate $r_{in} = N/D$ and reduce to lowest terms $m/n$. Return the value of $m+n$.


Question: Find the number of integer pairs $(x, y)$ such that $-100 \le x \le 100$, $-100 \le y \le 100$, and $4x - 3y = 0$.


Question: Calculate the sum of the fractions $\frac{23}{8}$ and $\frac{123}{8}$, simplify it to $\frac{m}{n}$, and report $m$ and $n$.


Question: The two possible values of $k$ are $k_1 = \frac{23}{8}$ and $k_2 = \frac{123}{8}$. Calculate their sum. Simplify the resulting fraction to the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Then report the value of $m+n$.


Question: Solve the equation $|36.5 - 4k| = 25$ for real $k$. Return the two possible values of $k$.


Question: In a trapezoid, the area is 72 and the height is 6. The parallel sides have lengths $r$ and $s$. Using the formula Area $= \frac{r+s}{2} \times \text{height}$, calculate the value of $r+s$.


Question: In an isosceles trapezoid with an inscribed circle of radius 3, what is the height?


Question: Find the number of integer pairs $(x, y)$ satisfying $3x + 2y = 0$ subject to the constraints $-100 \le x \le 100$ and $-100 \le y \le 100$.


Question: Evaluate the expression $r_{in} = \frac{45696}{11} \cdot \frac{1389920/1390041}{11/1179 + 1389920/1390041}$, simplify to the form $m/n$ where $m$ and $n$ are relatively prime positive integers, and return the value $m+n$.


Question: Compute the simplified fraction for the probability 1920/10395


Question: In an isosceles trapezoid with an inscribed circle of radius 3 tangent to the parallel bases, what is the height of the trapezoid?


Question: What is the value of 2304 minus 2025?


Question: Using the values $h_b = 45696/11$, $\sin(B/2) = 11/1179$, and $\cos^2(B/2) = 1389920/1390041$, calculate $r_{in} = h_b \frac{\cos^2(B/2)}{\sin(B/2) + \cos^2(B/2)}$. Express the resulting value as a simplified fraction $m/n$ where $m, n$ are relatively prime positive integers, and find the integer sum $m+n$.


Question: Solve for real $k$ given that the distance from point $(25, 20)$ to the line $8x - 6y - (8k + 7) = 0$ is equal to $5$. That is, solve $\frac{|8(25) - 6(20) - (8k + 7)|}{\sqrt{8^2 + (-6)^2}} = 5$.


Question: Let $k$ be a real number. Determine the Cartesian equation of the set of points $z=x+iy$ ($x, y \in \mathbb{R}$) satisfying $|z-(k+4)|=|z-(k+3i)|$. Express the result in the linear form $Ax+By=C$ where coefficients $A, B, C$ depend on $k$.


Question: What is 2304 minus 2025?


Question: Convert the base-$b$ representation '97' into an algebraic expression in terms of $b$ using place values, assuming the digit 9 is in the $b^1$ position and the digit 7 is in the $b^0$ position.


Question: Calculate the sum a+b+c given a=3, b=57, c=2.


Question: Calculate the sum of all real numbers k satisfying the equation |73 - 8k| = 50.


Question: Factor the quadratic expression $12x^2 - xy - 6y^2$ into two linear factors $(Ax + By)(Cx + Dy)$ and list the resulting linear equations equal to zero.


Question: What is the value of the number represented as $17$ in base $b$ expressed as a linear function of $b$?


Question: Calculate the number of integer pairs $(x, y)$ such that $-100 \le x \le 100$, $-100 \le y \le 100$, and $4x = 3y$.


Question: Find the number of integer pairs $(x, y)$ satisfying $3x = -2y$ subject to $-100 \le x \le 100$ and $-100 \le y \le 100$.


Question: Decompose the polynomial expression $12x^2 - xy - 6y^2$ into a product of two linear factors with integer coefficients of the form $(ax + by)(cx + dy)$. Provide the values of $a, b, c, d$ or the factors directly.


Question: Find the number of integer pairs $(x,y)$ such that $3x + 2y = 0$ and $-100 \le x, y \le 100$. Express your answer as the total count of such pairs.


Question: Given the circle center $C_1(25, 20)$ with radius 5, and the line being the perpendicular bisector of the segment connecting $(4+k, 0)$ and $(k, 3)$, find all real values of $k$ such that the distance from $(25, 20)$ to this line is equal to 5.


Question: For an isosceles tangential trapezoid where the sum of parallel sides r+s = 24 and the height is 6, calculate the value of r^2 + s^2.


Question: In an isosceles trapezoid with parallel sides of lengths $r$ and $s$, the height is 6 and the area is 72. Using the area formula $\text{Area} = \frac{1}{2}(r+s) \times \text{height}$, calculate the value of the sum $r+s$.


Question: Let $z = x + iy$ and $k$ be a real number. Simplify the equation $|z - (4+k)| = |z - (3i+k)|$ to find the linear relationship between $x$ and $y$. Express the final result in the form $A(x, y) + B(k)x + D = 0$ or similar linear format $Ax + By + C = 0$ where coefficients depend on $k$.


Question: In a trapezoid circumscribed about a circle of radius 3, what is the height?


Question: In the domain where $x$ and $y$ are integers such that $-100 \le x \le 100$ and $-100 \le y \le 100$, find the number of ordered pairs $(x,y)$ that satisfy the equation $4x - 3y = 0$.


Question: Calculate the value of $r^2+s^2$ using the algebraic identity $r^2+s^2 = \frac{(r+s)^2 + (r-s)^2}{2}$, given $r+s=24$ and $(r-s)^2=432$.


Question: Triangle ABC has vertices A, B, C. Points D, E lie on AB such that AD=4, DE=16. Points F, G lie on AC such that AF=13, FG=52. The quadrilateral DEGF (with vertices D, E, G, F in that order along the triangle sides) has area 288. Calculate the value of sin(angle BAC).


Question: Multiply 144 by 16.


Question: Calculate the sum of all real values of $k$ that satisfy the equation $(8k+73)^2 = 100(2k^2 + 14k + 25)$.


Question: What is the difference between 2304 and 2025?


Question: Find all integer bases $b > 9$ such that $(b + 7)$ is a positive divisor of $56$.


Question: Calculate $51 + 67 - 1$.


Question: Calculate the maximum value of $x^2+y^2+z^2$ for positive real numbers $x, y, z$ satisfying $xy+yz+zx=27$ and $xyz=23$.


Question: Calculate the number of ways to partition the set $\{A, B, C, D, E, F, G, H, I, J, K, L\}$ into 6 unordered pairs.


Question: Find the value of $r^2 + s^2$ for an isosceles trapezoid where the parallel sides sum to 24, the height is 6, and the trapezoid has an inscribed circle (which implies the sum of the parallel sides equals the sum of the legs).


Question: What is the sum of the values 23/8 and 123/8?


Question: Calculate the sum $2\cdot 7 + 3\cdot 4 + 5\cdot 1 + 7\cdot 1$.


Question: Find the two linear factors $(ax+by)(cx+dy)$ such that $(ax+by)(cx+dy) = 12x^2-xy-6y^2$.


Question: In the case where $4x=3y$, substitute $x=3k$ and $y=4k$ where $k$ is an integer. Determine the number of distinct integer values of $k$ such that both $-100 \le 3k \le 100$ and $-100 \le 4k \le 100$.


Question: Find the number of integer pairs $(x,y)$ that satisfy the equation $4x - 3y = 0$ while also satisfying the condition $-100 \le x \le 100$ and $-100 \le y \le 100$.


Question: Factor the homogeneous quadratic expression $12x^2 - xy - 6y^2$ into the product of two linear factors of the form $(Ax + By)(Cx + Dy)$. List the resulting two linear equations $Ax + By = 0$ and $Cx + Dy = 0$.


Question: What is 1 + 27 + 403/3?


Question: What is the number of integer pairs (x, y) satisfying 3x = -2y where -100 ≤ x, y ≤ 100?


Question: Calculate (r-s)^2 for an isosceles trapezoid with height h=6, where the sum of the parallel sides is r+s=24, and an inscribed circle is tangent to all four sides.


Question: How many integer pairs $(x,y)$ satisfy the equation $3x + 2y = 0$ subject to the constraints $-100 \le x \le 100$ and $-100 \le y \le 100$?


Question: Calculate 25 + 193 + 18


Question: Calculate 8 times 288, then subtract 2025 from the result.


Question: What is the sum of the integers 1, 11, and 37?


Question: Calculate the total number of distinct integer pairs by performing the operation $67 + 51 - 1$.


Question: In triangle ABC, D and E lie on side AB such that AD=4, DE=16, EB=8. F and G lie on side AC such that AF=13, FG=52, GC=26. We are given that Area(DEGF) = 288. Note that Area(DEGF) = Area(AEG) - Area(ADF). Calculate the area of triangle ABC.


Question: Factor the quadratic expression $12x^{2}-xy-6y^{2}$ into a product of two linear factors in $x$ and $y$. Provide the factors $(Ax + By)(Cx + Dy) = 0$.


Question: Identify all subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that contain exactly 4 distinct digits and have a sum of 18. For each subset identified, please provide the count of even numbers included in it.


Question: Find the inradius $r_{in}$ of triangle ABC, given that 8 circles of radius 34 and 2024 circles of radius 1 can be arranged sequentially and tangent to sides AB and BC in the same manner within the triangle.


Question: Calculate the difference between 2304 and 2025.


Question: Calculate the numerical value of 3 * (-2 + 3) * ((-2)^2 + 9).


Question: Calculate \(3^{23} \pmod{1000}\).


Question: Find the number of integer values of $k$ such that the pair defined by $x=-2k$ and $y=3k$ satisfies the condition that both $x$ and $y$ are integers within the inclusive range $[-100, 100]$. Report only the count of such integers $k$.


Question: What is the sum of the positive integers 1, 11, and 37?


Question: In the complex plane, let $z = x+iy$. Simplify the equation $|z-(4+k)|=|z-(3i+k)|$ where $k$ is a real number to find its Cartesian line equation in the form $Ax+By=C$.


Question: Find the number of integer pairs $(x, y)$ such that $x$ and $y$ are integers between $-100$ and $100$ inclusive, and satisfy the equation $4x-3y=0$.


Question: Calculate the value of $3(-2 + 3)((-2)^2 + 9)$.


Question: What is the sum of 1 + 11 + 37?


Question: What is the sum of 1, 11, and 37?


Question: What is the value of 8 * 4! * 2 * 3!?


Question: What are the positive divisors of 39 that are greater than or equal to 3?


Question: Calculate the value of the expression 3(-2+3)((-2)^2+9).


Question: Given the point (25, 20) and the line 8x - 6y - (7 + 8k) = 0, find the real values of k such that the perpendicular distance from the point to the line equals 5.


Question: What is the number of subsets of size 4 that can be formed from the set {1, 2, 3, 4, 5, 6, 7, 8} such that the sum of the elements in the subset is 18?


Question: Verify the algebraic simplification of x-yz < y-zx on the plane x+y+z=75. Show that this inequality is equivalent to (x-y)(1+z) < 0.


Question: Find all positive integer divisors of 39.


Question: Find the number of subsets O of {1,2,3,4,5,6,7,8} such that |O|=4 and sum(O)=18. Calculate the sum of the number of even digits in O across all such subsets.


Question: What is the value of 9 factorial?


Question: Calculate the area of a triangle with base length 39 and height 24.


Question: What is the sum of 1, 11, and 37?


Question: Calculate the value of the expression $ 3(-2 + 3)((-2)^2 + 9) $.


Question: Let $n = k - 2$. Substitute this into the expression $3(n + 3)(n^2 + 9)$. Expand the expression as a polynomial in $k$. Find the constant term $C$ (the remainder when divided by $k$). Report the value of $C$.


Question: List the positive integer divisors of 39.


Question: Solve for $n$ in the equation $n + 2 = d$ for each divisor $d \in \{1, 3, 13, 39\}$. Filter the results to include only positive integers (where $n > 0$). Finally, calculate the sum of these positive integer values of $n$.


Question: What is 39 multiplied by 12?


Question: Substitute $n = k-2$ into $3(n+3)(n^2+9)$. Expand the polynomial $3(k+1)(k^2-4k+13)$ fully. Identify the constant term resulting from this expansion, which represents the divisibility constraint $k \mid C$.


Question: Calculate the remainder when $3^{11}$ is divided by $1000$.


Question: Find all positive integers $n$ such that $n+2$ is a divisor of 39 and $n \ge 1$. List these values.


Question: Solve the quadratic equation $3y^2 + 35y + 36 = 0$ and the quadratic equation $9y^2 + 25y + 12 = 0$. Return the real roots.


Question: Calculate the sum of all positive integers $n$ such that $n = k - 2$ where $k$ is a divisor of 39 and $k \ge 3$.


Question: What is the value of $39 \times 12$?


Question: Find the value of the product $P_{log} = \prod_{k=4}^{63} \frac{\ln(k+1)}{\ln k}$ where $\ln$ denotes the natural logarithm.


Question: Calculate the sum $S = \sum_{A \subseteq \{1,\dots,8\}, |A|=4, \sum A=18} |A \cap \{2,4,6,8\}|$.


Question: Points $A, B, C, D, E, F$ lie in a straight line in that order. Set the coordinate of $A$ to 0. Given the following segment lengths: $AC=26$, $BD=22$, $CE=31$, $DF=33$, $AF=73$. Compute the numeric coordinates for points $B, C, D, E, F$ on this line.


Question: Points A, B, C, D, E, F lie on a straight line in that order. Known lengths are AC = 26, BD = 22, CE = 31, DF = 33, and AF = 73. Calculate the length of the segment BE.


Question: What is the number of subsets of {1, 2, 3, 4, 5, 6, 7, 8} that contain exactly 4 elements and sum to 18?


Question: Calculate $y = \left(\frac{\sqrt{19}-\sqrt{3}}{2}\right)^2 - 4$. Show that it simplifies to $\frac{3-\sqrt{57}}{2}$. Given this matches the form $\frac{a-\sqrt{b}}{c}$ with $a=3, b=57, c=2$ (checking $\gcd(3,2)=1$), compute the integer sum $a+b+c$.


Question: Given points C and D lie on a line with coordinates 26 and 40 respectively. A point G forms right triangles with their projections on the line such that CG = 40 and DG = 30. Calculate the perpendicular distance $h$ from point G to the line containing C and D.


Question: Calculate the area of a triangle with base length 39 and height length 24.


Question: What is the prime factorization of the integer $362880 \times 56 \times 46656$ in the form $2^a \cdot 3^b \cdot 5^c \cdot 7^d$? Please report the specific values of the exponents a, b, c, and d.


Question: Solve the system $u=2v$ and $u^2+v^2=36$ to find $v^2$, then compute the Area $= 8v^2$ as an irreducible fraction $m/n$ and report $m+n$.


Question: Calculate the height of point G from the line using the distances CG = 40, DG = 30, and the known positions C = 26, D = 40. Then calculate the area of triangle BGE with base BE = 39.


Question: Calculate the value of the rational product R = prod_{k=4}^{63} ((k-1)(k+1)/((k-2)(k+2))).


Question: Calculate $N_6 = \frac{5}{9} 3^{12}$ and report $N_6 \pmod{1000}$.


Question: Calculate the value of $R = \prod_{k=4}^{63} \frac{k^2-1}{k^2-4}$.


Question: Calculate 72 + 2*24 + 3*48


Question: Given points A,B,C,D,E,F are collinear in order. With AC=26, CE=31, DF=33, AF=73 (and thus coordinates determined relative to A), what are the coordinates of C and D relative to A? Additionally, given CG=40 and DG=30 where G is not on the line, what is the perpendicular distance from G to the line?


Question: Count the number of subsets of size 4 from {1, 2, 3, 4, 5, 6, 7, 8} that sum to 18.


Question: Assign coordinate 0 to point A on a line. Given points B, C, D, E, F lie in order with A and satisfy AC=26, BD=22, CE=31, DF=33, and AF=73. Calculate the coordinates of points B, C, D, E, and F.


Question: Consider triangle formed by points C, D on a straight line and point G not on the line. The length of the base segment CD is 14. The lengths of the other two sides are CG = 40 and DG = 30. Calculate the perpendicular distance from point G to the line passing through C and D.


Question: Evaluate the sum $25 + 16 + 16 + 25$.


Question: Given points C and D lie on a straight line with horizontal coordinates $x_C=26$ and $x_D=40$. A point G lies off the line such that the distance $CG = 40$ and $DG = 30$. Calculate the perpendicular distance $h$ from point G to the line containing C and D.


Question: What is the sum 1349/6?


Question: In a geometric configuration where points A, B, C, D, E, F lie on a straight line in that order, the following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, and AF = 73. Calculate the length of the segment BE.


Question: No query needed, finalize answer.


Question: What is 507 plus 2?


Question: Given six points $ A, B, C, D, E, F $ lying on a straight line in that order, with lengths $ AC = 26 $, $ BD = 22 $, $ CE = 31 $, $ DF = 33 $, and $ AF = 73 $. Please calculate the lengths of segments $ CD $ and $ BE $.


Question: Given a triangle with side lengths 14, 40, and 30. If the base of the triangle is 14, find the length of the altitude drawn from the opposite vertex to this base.


Question: Simplify the expression $\frac{\log_k(5^{k^2-1})}{\log_{k+1}(5^{k^2-4})}$ using the change-of-base formula and power rules to express it as a product of a rational function of k and a ratio of natural logarithms.


Question: Calculate the area of the equilateral triangle with vertices $(75,0,0)$, $(0,75,0)$, and $(0,0,75)$ in 3D space.


Question: Calculate the values of $K(0,0) = \binom{2}{0}\binom{2}{0} + \binom{2}{1}\binom{2}{1}$, $K(0,1) = \binom{2}{0}\binom{2}{1} + \binom{2}{1}\binom{2}{2}$, $K(1,0) = \binom{2}{1}\binom{2}{0} + \binom{2}{2}\binom{2}{1}$, and $K(1,1) = \binom{2}{1}\binom{2}{1} + \binom{2}{2}\binom{2}{2}$.


Question: Given $2025 = 3^4 \times 5^2$, how many positive integer divisors does 2025 have?


Question: How many subsets of 4 distinct digits from {1, 2, 3, 4, 5, 6, 7, 8} sum to 18?


Question: Calculate the value of the rational telescoping product: $\prod_{k=4}^{63} \frac{k^2-1}{k^2-4}$.


Question:
Compute the value of the following summation:
$$ \sum_{x_1, x_2, x_3, x_4 \in \{0, 1\}} \left[ \binom{2}{x_1+x_4} \binom{2}{x_2+x_4} \binom{2}{x_1+x_3} \binom{2}{x_2+x_3} \right] $$
where $x_i$ are binary variables (0 or 1) and $\binom{n}{k}$ is the binomial coefficient defined for integer $n \ge 0, k \ge 0$. Note that if $x_i+x_j > 2$, treat $\binom{2}{x_i+x_j} = 0$ (though sum $\le 2$ here).


Question:
Given $u^2 + v^2 = 36$ and $u = 2v$:
1. Calculate $v^2$.
2. Calculate the Area of Rectangle $EFGH$ using Formula Area $= 8v^2$.
Report the Area.


Question: Given a rectangle with width w and height h such that h = 2w. Its vertices lie on a circle of radius 6, so w² + h² = 144. Calculate the area of the rectangle as a simplified fraction m/n (where m and n are coprime), and find the sum m + n.


Question: CALCULATE: 72 + 2*24 + 3*72


Question: Let $n = 2025$. Calculate the following values: (1) The total number of positive divisors of $n$ (let this be $N_{tot}$). (2) The number of positive divisors of $n$ divisible by $3^4$ (let this be $N_{3^4}$). (3) The number of positive divisors of $n$ divisible by $5^2$ (let this be $N_{5^2}$). (4) The number of positive divisors of $n$ divisible by both $3^4$ and $5^2$ (let this be $N_{both}$).


Question: In the plane x+y+z=75, find the point(s) where the boundaries x-yz=y-zx and y-zx=z-xy simultaneously hold, i.e., solve x-yz=y-zx and y-zx=z-xy along with x+y+z=75


Question: Given that points A, B, C, D, E, F lie on a line in that order, with AC = 26, DF = 33, and AF = 73, calculate the length of segment CD.


Question: Simplify the general term $T_k = \frac{\log_k(5^{k^2-1})}{\log_{k+1}(5^{k^2-4})}$ using logarithm laws ($\log_a x^y = y \log_a x$ and $\log_a b = \frac{\ln b}{\ln a}$). Provide the resulting expression as a product of a rational function of k and a ratio of logarithms.


Question: Calculate the sum: $9 + 252 + 1260 + 1260 + 126$.


Question: Calculate the area of triangle BGE given its base along the line is 39 and the height (perpendicular distance from G to the line) is 24.


Question: Evaluate the sum $S = \sum_{x,y,z,w \in \{0,1\}} \binom{2}{2-x-y} \binom{2}{2-x-w} \binom{2}{2-y-z} \binom{2}{2-z-w}$, where $x$ represents the color of the first internal vertical segment, $y$ the second, $z$ the third, and $w$ the fourth, corresponding to the four squares' internal edge constraints.


Question: Calculate the length of segment BE. Given that points A, B, C, D, E, F lie on a line in that order, and the lengths AC = 26, BD = 22, CE = 31, and CD = 14. Determine the distance between point B and point E.


Question: Calculate the area of quadrilateral AFNBCEM based on the decomposition Area = Area(ABC) + Area(BNE) + Area(CMF).

\text{Area}(ABC) + \text{Area}(BNE) + \text{Area}(CMF)


Question: Compute the value of the product \prod_{k=4}^{63} \frac{\ln(k+1)}{\ln k}.


Question: Given $k = 2l$ and $k^2 + l^2 = 36$, find the area of the rectangle defined by half-width $k$ and half-height $l$ (which is $4kl$). Express this area as an irreducible fraction $m/n$ and compute $m+n$.


Question: Verify that $c=185$ is square-free and compute $a+b+c+d$.


Question: Apply the law $\log_a(b^c) = c\log_a b$ and $\log_a b = \frac{\ln b}{\ln a}$ to the general term $\frac{\log_k(5^{k^2-1})}{\log_{k+1}(5^{k^2-4})}$. Express the result in the form $R_k \cdot L_k$ where $R_k$ is a rational function of $k$ and $L_k$ involves $\ln k$. Report the expression for $R_k$.


Question: What is the number of positive integer divisors of 2025?


Question: Calculate the value of the sum $N = \sum_{(a,b,c,d) \in \{0,1\}^4} 2^{\mathbb{I}(a \neq b) + \mathbb{I}(b \neq c) + \mathbb{I}(c \neq d) + \mathbb{I}(d \neq a)}$, where $\mathbb{I}(\cdot)$ is the indicator function. Return the exact integer.


Question: Calculate the sum of the positive integers 109 and 128.


Question: In the collinear arrangement of points A, B, C, D, E, F where AC=26, BD=22, CE=31, DF=33, and AF=73, find the length of segment BE.


Question: Calculate the value of the numerator count: $2^{15} - 2^{12} - 2^{10} + 2^8$. Provide the exact integer result.


Question: Calculate the perpendicular distance from point G to the line containing A,B,C,D,E,F, given that CG=40, DG=30, and the distance between C and D is 14.


Question: Calculate the exact value of the product of the rational factors: $\prod_{k=4}^{63} \frac{k^2-1}{k^2-4}$.


Question: In a rectangular coordinate system, let a rectangle have width $w$ and height $h$, centered at $(0,6)$, aligned with axes. Its top side $GH$ lies on $y=6+h/2$ and right side $FG$ lies on $x=w/2$. Given points $C(0,30)$ and $D(12,6)$, calculate the value of $h/w$ if the area of triangle $DGF$ (vertices $D, G, F$) equals the area of triangle $CHG$ (vertices $C, H, G$).


Question: What is the prime factorization of 2025? How many positive integer divisors does it have? Furthermore, how many of these divisors have the exponent of 3 equal to 4 (maximum possible), and how many have the exponent of 5 equal to 2 (maximum possible)?


Question: Calculate the value of $2^{15} - 2^{12} - 2^{10} + 2^8$. Divide this value by $2^{15}$ to find the probability in simplest form $m/n$. Find the sum $m+n$.


Question: Calculate the sum of 495 + 12870 + 45045 + 45045 + 12870 and report the remainder when divided by 1000.


Question: Find the prime factorization of 2025 and calculate the number of its positive integer divisors.


Question: In triangle ABC, D and E lie on AB with AD=4 and AE=20. F and G lie on AC with AF=13 and AG=65. The area of quadrilateral DEGF is 288. Calculate the area of triangle ABC.


Question: Calculate the sum S over all binary tuples $(u,v,w,z) \in \{0,1\}^4$ of the product $K(u,w) K(v,w) K(u,z) K(v,z)$, where $K(x,y) = 2 - \frac{x+y}{x+y}$ if $x+y \neq 0$? No, use the integer definition: K(x,y) = 1 if x+y is even, 2 if x+y is odd.


Question: Solve the equation $3x^4 + 2\sqrt{3}x^3 - 25x^2 - 6\sqrt{3}x + 40 = 0$ for the unique real root $x$ satisfying $0 < x < 2$. Report the value of $x$.


Question: For $u \in (0, 10\pi)$, how many solutions does the equation $\sin(u) = 0$ have?


Question: Solve the system of equations: $a^2 + b^2 = 36$ and $a = 2b$. Find the values of $a$ and $b$, then compute the Area of the rectangle $EFGH$ defined as $4ab$. Express the area as a simplified fraction $m/n$. Calculate $m+n$.


Question: Is 610 the correct number of successes?


Question: Evaluate the rational part of the product: \prod_{k=4}^{63} \frac{k^2-1}{k^2-4}.


Question: Determine $n$, the number of real values of $x$ in the interval $(0, 2\pi)$ that satisfy the equation $\sin(7\pi \cdot \sin(5x)) = 0$. Let $u = 5x$, so $x \in (0, 2\pi)$ maps to $u \in (0, 10\pi)$. The condition is equivalent to $\sin(u) = \frac{k}{7}$ for some integer $k$. Identify the valid range for $k$ (based on the bounds of sine), and for each integer $k$, determine the number of solutions for $u$ in $(0, 10\pi)$. Sum these counts to report the value of $n$.


Question: CALCULATE the expected number of regions.


Question: Set up a coordinate system where circle $\omega_2$ has center at $(0,0)$ and radius 15. Let point $B$ be at $(-15, 0)$ and $C$ at $(15, 0)$ such that $\overline{BC}$ is a diameter. Circle $\omega_1$ has radius 6 and is internally tangent to $\omega_2$ at point $B$. Find the coordinates of the center $A$ of $\omega_1$. Next, find the y-coordinate(s) of a point $D$ on $\omega_2$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$. Specifically, calculate the absolute value of the y-coordinate of $D$. Also, assuming rectangle $EFGH$ is inscribed in $\omega_1$ with sides parallel to the


Question: Analyze the inequalities $(x-y)(1+z)<0$ and $(y-z)(1+x)<0$ on the plane $x+y+z=75$. Determine the vertices of the finite region defined by $x<y<z$ and $x,y,z>-1$, and calculate its area.


Question: I need to recalculate the total intersections. New-Chord-Initial: 25 * 4/3 = 100/3. New-Chord-New: 300 * 8/9 = 800/3. Sum Intersections = 900/3 = 300. Regions = 1 + 27 + 300 = 328.


Question: What is the prime factorization of 2025 and what is the total number of its positive integer divisors?


Question: Calculate the sum of the following integers: 2, 4, 8, 16, 2, 64, 2, 0, 8, 4, 2, 1.


Question: In the interval $0 < u < 10\pi$, find the number of solutions to $\sin(u) = 0$.


Question: In the function $f(x)$ with period 4 defined by $f(x)=x$ on $[-1,1)$ and $f(x)=2-x$ on $[1,3)$, consider the intersections with $x=34y^2$ for $x \in [0, 34]$. Focus on the Rising segments where $f(x) = x - 4k$ for integer $k$. Specifically, solve the system for $k \in \{0, 1, \dots, 8\}$, keeping only solutions where $x \in [4k-1, 4k+1)$ and $x \le 34$. Calculate the sum of the valid $y$-coordinates $f(x)$ found in these Rising intervals.


Question: Calculate the sum $S = \sum_{a,b,c,d \in \{0,1\}} \binom{2}{a+b} \cdot \binom{2}{b+c} \cdot \binom{2}{a+d} \cdot \binom{2}{c+d}$ where $\binom{n}{k}$ is the binomial coefficient.


Question: Calculate the binomial coefficients $\binom{16}{8}$, $\binom{13}{8}$, and $\binom{10}{8}$. Then compute the value of $N = \binom{16}{8} - 9\binom{13}{8} + 36\binom{10}{8}$ and report the remainder when $N$ is divided by 1000.


Question: CALCULATE the sum of the products over all $A, B, C, D \in \{0, 1\}$ of $\binom{2}{A+B} \binom{2}{A+C} \binom{2}{B+D} \binom{2}{C+D}$, where the binomial coefficient $\binom{n}{k}$ is defined as 0 if $k < 0$ or $k > n$.


Question: Calculate the value of $2^{15} - 2^{12} - 2^{10} + 2^8$. Form the probability fraction $P = \frac{\text{result}}{2^{15}}$, simplify it to lowest terms $\frac{m}{n}$ where $\gcd(m,n)=1$, and compute the sum $m+n$. Provide the final answer in the format \boxed{result}.


Question: Find the remainder when 2907 is divided by 1000.


Question: In a rectangle centered at $(0,0)$ with vertices $(\pm k, \pm l)$, consider $C(-24,0)$ and $D(0,-12)$. Using the coordinates, express the area of $\triangle DGF$ (with vertices $D, G, F$) and $\triangle CHG$ (with vertices $C, H, G$). Solve the equation Area($\triangle DGF$) = Area($\triangle CHG$) for the ratio $k/l$.


Question: In the interval $0 < u < 10\pi$, find the number of solutions to $\sin(u) = 1$. Find the number of solutions to $\sin(u) = -1$. Sum these two counts to obtain the value of $t$ (tangency count).


Question: Given a triangle with area 1 and an included angle $\theta$ such that $\cos \theta = \frac{12}{13}$, find the product of the lengths of the sides adjacent to $\theta$.


Question: Calculate the sum $S = \binom{8}{0}\binom{9}{8} + \binom{7}{1}\binom{9}{7} + \binom{6}{2}\binom{9}{6} + \binom{5}{3}\binom{9}{5} + \binom{4}{4}\binom{9}{4}$. Find $S \pmod{1000}$.


Question: In the interval $u \in (0, 10\pi)$, find the total number of solutions $n$ to the equation $\sin(u) = k/7$, where $k$ is an integer satisfying $|k| \le 7$. Sum the count of solutions across all such $k$.


Question: In Case 1, the intersection equation is $34y^2 - y - 4k = 0$ for integer $k$. Given that valid solutions require $y \in [-1, 1)$, determine the sum of all such valid roots $y$ across all applicable integers $k$.


Question: Calculate the sum of $M_k$ for $k=1$ to $12$, where $M_k$ is the number of perfect matchings of length $k$ segments on a regular 24-gon. Rules: If $k=12$, $M_k=1$. If $k < 12$, let $g=\gcd(k, 24)$. If $24/g$ is even, $M_k=2^g$. If $24/g$ is odd, $M_k=0$. Return the total sum.


Question: Determine the implicit equation $F(x,y)=0$ for the parabola formed by rotating $y=x^{2}-4$ by $60^{\circ}$ counterclockwise around the origin. To find this, let $(x_0, y_0)$ be a point on the original parabola and $(x, y)$ be its image. The relationship is $x_0 = x \cos(-60^{\circ}) - y \sin(-60^{\circ}) = \frac{1}{2}x + \frac{\sqrt{3}}{2}y$ and $y_0 = x \sin(-60^{\circ}) + y \cos(-60^{\circ}) = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y$. Substitute $x_0, y_0$ into $y_0 = x_0^2 - 4$ to obtain $F(x,y) = 0$ in simplest form.


Question: Calculate the value of the product $L = \prod_{k=4}^{63} \frac{\ln(k+1)}{\ln(k)}$.


Question: Calculate the sum of all valid roots y from the equation $34y^2 + y - (4k+2) = 0$ where k is an integer and $y \in [-1, 1]$. Identify the valid range of k and the total sum.


Question:
Please calculate the total number of ways to partition the 24 vertices of a regular 24-gon into 12 pairs of equal length. Use the following rule for each step size $s \in \{1, \dots, 12\}$:
1. Compute $g = \gcd(s, 24)$ and $N = 24/g$.
2. If $N$ is odd, the contribution is 0.
3. If $N = 2$, the contribution is 1.
4. If $N > 2$ and even, the contribution is $2^g$.
Sum the contributions for all $s$ from 1 to 12.


Question: Calculate the number of values $x$ in the interval $0 < x < 2\pi$ such that $f(x) = \sin(7\pi \cdot \sin(5x)) = 0$.


Question: Simplify the term $\frac{\log_k(5^{k^2-1})}{\log_{k+1}(5^{k^2-4})}$ using $\log(a^b)=b\log a$ and change of base formulas, expressing it in terms of $(k-1)(k+1)/(k-2)(k+2)$ and $\frac{\ln(k+1)}{\ln k}$.


Question:
Task: Given the problem analysis summarized below, calculate the final integer answer.
Context:
Vertices: 24.
Segments: 12 pairs, equal chord length.
Methodology per chord step $m \in \{1, \dots, 12\}$:
1. Calculate $g = \gcd(m, 24)$.
2. Calculate cycle length $L = 24/g$.
3. If $L$ is odd, count = 0.
4. If $L$ is even, count = $2^g$.
5. Sum the counts for all $m$.
Execution: Perform the summation for $m=1$ to $12$.
Answer format: The single integer sum.


Question: Let $g_{opt}(n)$ be the minimum number of 1 and 10 cent coins needed to make $n$ cents. Compare the coin count of the greedy algorithm for $N = 25k + r$ (where $k \ge 1$) against a solution using $k-1$ twenty-five cent coins. Specifically, find the values of $r \in \{0, 1, \dots, 24\}$ where $g_{opt}(r) + 1 > g_{opt}(r+25) + 1$? No, find where the greedy count exceeds the optimal count using $k-1$ twenty-five coins. That is, for fixed $k \ge 1$, does the greedy strategy yield fewer coins? Or more? Output the set of $r$ where Greedy is suboptimal.


Question: Calculate the value of $2^{15} - 2^{12} - 2^{10} + 2^8$. Report the result as an integer.


Question: The number of subsets of $\{1, 2, \dots, 16\}$ with cardinality 8 is $N$. A subset is valid if no three elements form a sequence of consecutive integers (e.g. $\{1,2,3\}$ is invalid). Calculate $N \pmod{1000}$.


Question: Calculate the value of the rational product R = \prod_{k=4}^{63} \frac{(k-1)(k+1)}{(k-2)(k+2)}


Question: What is the prime factorization of 2025, and what is the number of positive integer divisors of 2025?


Question: Compute S_4 (m_4+n_4) mod 1000 to verify the cycle.


Question: In the interval $0 < u < 10\pi$, find the total number of solutions to $\sin(u) = k/7$ for all integers $k$ satisfying $|k| \le 7$. This total should represent $n$.


Question: Find the positive real solution $S$ to the equation $S + 9\sqrt{S^2 - 20} = 20$.