Question: Calculate $\sin(\theta)$ where $\theta = \angle A_iA_1A_{i+1}$ given $\cos(\theta) = \frac{12}{13}$.


Question: Perform the algebraic elimination of $x$ using the system: 1) $x^2 = y + 4$ and 2) $x^2 + 2\sqrt{3}xy + 3y^2 + 2\sqrt{3}x - 2y - 16 = 0$. Substitute (1) into (2), collect all terms containing $x$ on one side, isolate $x$ (linear in $x$), square both sides to eliminate $x$, and substitute $x^2 = y + 4$ once more to clear $x^2$. Simplify the expression to obtain a polynomial equation in the form $P(y) = 0$. Return the polynomial $P(y)$. Do not solve for $y$ yet.


Question: Reduce the fraction $\frac{27904}{32768}$ to lowest terms $\frac{m}{n}$, ensure $m$ and $n$ are relatively prime, and calculate the value of $m+n$.


Question: Calculate the value of $\binom{16}{8} - 9\binom{13}{8} + 36\binom{10}{8}$.


Question: What is $673 + 575 \pmod{1000}$?


Question: Compute the value of $S = \sum_{a,b,c,d \in \{0,1\}} f(a+d) \cdot f(b+d) \cdot f(a+c) \cdot f(b+c)$, where $f(k) = 1$ if $k=0$ or $k=2$, and $f(k) = 2$ if $k=1$. Here $a,b,c,d$ represent the 4 internal edges of the grid.


Question: Calculate the value of the sum $\binom{9}{4}\binom{4}{4} + \binom{9}{5}\binom{5}{3} + \binom{9}{6}\binom{6}{2} + \binom{9}{7}\binom{7}{1} + \binom{9}{8}\binom{8}{0}$. Report the resulting integer $N$.


Question: Find the number of distinct real solutions $x \in (0, 2\pi)$ to the equation $\sin(5x) = k/7$ for each integer $k \in \{-7, -6, \dots, 6, 7\}$, and return the sum of these counts as $n$.


Question: Solve the equation $X + 9\sqrt{X^2 - 20} = 20$ for $X$, where $X$ represents the sum of two segment lengths $u_2 + u_{11}$. Provide the exact value of $X$ in simplest radical form.


Question: Solve the polynomial equation $3x^4 + 2\sqrt{3}x^3 - 25x^2 - 6\sqrt{3}x + 40 = 0$ for the unique real root $x$ that lies strictly between 0 and 2.


Question: Evaluate the expression $72 + 2 \cdot 168 + 3 \cdot 72$.


Question: Let $S$ be the positive real number satisfying the equation $S + 9\sqrt{S^2 - 20} = 20$. Compute $S$ and express it in the form $\frac{m\sqrt{n} - p}{q}$ for positive integers $m, n, p, q$ with $n$ squarefree. Determine the value of $m+n+p+q$.


Question: Write a Python script that counts the number of binary strings of length 16 with exactly eight 1s that do not contain the substring "111". Return the count.


Question: In triangle ABC with A at origin, B on x-axis, C on y-axis. Given K, L inside such that AK=AL=KL=14 and BK=14, CL=14. Let alpha = angle KAB. Show that c = 28 cos(alpha) and b = 28 sin(alpha+60 degrees).

c = 28 \cos \alpha, b = 28 \sin(\alpha + 60^\circ)


Question: Consider the quartic equation $9y^4 - 18y^3 - 143y^2 - 84y + 96 = 0$. Identify the unique real root $y$ located in the interval $[-4, 0)$ (corresponding to the fourth quadrant intersection). Express this root as $y = \frac{a - \sqrt{b}}{c}$ where $a, b,$ and $c$ are positive integers and $a$ and $c$ are relatively prime. Report the calculated value of the sum $a + b + c$.


Question: What is the area of quadrilateral BKLC given the conditions?


Question: In a triangle with area 1 and one angle $\theta$ such that $\cos \theta = 12/13$, find the product of the lengths of the two sides enclosing $\theta$.


Question: Assume $x_{2025}$ converges to the fixed point $1/2$. Calculate $m+n$ where $x_{2025} = 1/2 + 0 = 1/2$. Report $1+2 \pmod{1000}$.


Question:
For each $d$ in {1, ..., 12}: calculate $N(d)$ where $g=\gcd(d, 24)$ and $m=24/g$.
If $m$ is odd, $N(d)=0$.
If $m$ is even and $m=2$, $N(d)=1$.
If $m$ is even and $m>2$, $N(d)=2^g$.
Compute $S = \sum N(d)$ and return it.


Question: Given $\alpha + \delta = 30^\circ$, $\cos(2\alpha - 30^\circ) = \frac{55\sqrt{3}}{98}$, and Area($BKLC$) = $49\sqrt{3}(4 \cos^2 \alpha - 1)$, calculate the value of $n$.


Question: Determine the prime factorization of 2025. If $2025 = 3^a \cdot 5^b \cdots$, what are the values of $a$ and $b$? Also, calculate the total number of positive integer divisors of 2025, denoted as $|A|$.


Question: Calculate the sum of the positive roots 18, 72, and 98.


Question: Count the number of integers $N$ between 1 and 1000 inclusive such that $\lfloor N/25 \rfloor \ge 1$ and $N \pmod{25}$ is in the set $\{5, 6, 7, 8, 9, 15, 16, 17, 18, 19\}$. Please return the exact count.


Question: Compute 72 + 2*12 + 3*72.


Question: Compute the value of the probability $P = \frac{2^{15} - 2^{12} - 2^{10} + 2^8}{2^{15}}$ and simplify it to the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers.


Question: Verify the intersection point coordinates for the parabola $y=x^2-4$ rotated $60^\circ$ counterclockwise. Specifically, determine if the y-coordinate is $\frac{1-\sqrt{13}}{2}$.


Question:
Solve for $y$ given the system of equations:
1) $x^2 = y + 4$
2) $x^2 + 2\sqrt{3}xy + 3y^2 + 2\sqrt{3}x - 2y - 16 = 0$
Substitute $x^2$ from (1) into (2), rearrange to isolate terms with $x$, square to eliminate $x$, and substitute $x^2=y+4$ again. Derive the resulting polynomial equation solely in terms of $y$, then find the real root $y$ such that $y < 0$.


Question: Determine the set of integers $n \in \mathbb{Z}$ such that the interval representing the $n$-th period, $[4n-1, 4n+3)$, has a non-empty intersection with the interval $[0, 34]$.


Question: Given triangle $ABC$ with angles $84^\circ, 60^\circ, 36^\circ$ at $A, B, C$, let $\Omega$ be its Nine-Point Circle. Let $H$ be the foot of the altitude from $B$ to $AC$, and $J$ be the foot of the altitude from $C$ to $AB$. Both $H$ and $J$ lie on $\Omega$. Find the measure of the minor arc $HJ$.


Question:
In a right triangle $ABC$ with $\angle A = 90^\circ$ and hypotenuse $BC=38$, let $AB=b$ and $AC=c$. There exist points $K$ and $L$ inside $\triangle ABC$ such that $AK=BK=14$, $AL=CL=14$, and $KL=14$.
It can be shown that $\triangle AKL$ is equilateral. Using the coordinate geometry or trigonometry implied by these distance constraints, find the value of the product $bc$.


Question: Eliminate the variable $x$ from the system consisting of $y=x^2-4$ and $x^2 + 3y^2 + 2\sqrt{3}xy - 2y + 2\sqrt{3}x - 16 = 0$ to derive the resulting polynomial equation in $y$. Solve this polynomial equation to find all real roots. Identify the unique real root $y$ that satisfies the condition for a point in the fourth quadrant ($y < 0$).


Question: Find the number of values of $x$ in the interval $0 < x < 2\pi$ such that $\sin(7\pi \sin(5x)) = 0$. Report this number as $n$.


Question: In triangle ABC with angles 84, 60, 36, let D, E, F be midpoints of BC, AC, AB. The circumcircle of DEF intersects BD, AE, AF at G, H, J. Calculate the measure of the minor arc DE.


Question: CALCULATE: Let $S = x+y$ where $xy = 5.2$ and $S + 9\sqrt{S^2 - 20} = 20$. Solve for $S$ and express it in the form $\frac{m\sqrt{n} - p}{q}$ with $m,n,p,q$ positive integers and no common factor for $m,p,q$ and $n$ squarefree.


Question: In an 11-sided non-convex simple polygon $A_1A_2\ldots A_{11}$, let $r_i = |A_1A_i|$. We are given $r_i r_{i+1} = \frac{26}{5}$ for $i=2,\dots,10$, $\cos(\angle A_iA_1A_{i+1}) = \frac{12}{13}$, and the perimeter is 20. The perimeter constraint allows us to form an equation for $S = r_2 + r_{11}$. Calculate the value of $S$ exactly in the form $\frac{m\sqrt{n}-p}{q}$ where $n$ is squarefree and $gcd(m,p,q)=1$. Then determine $m+n+p+q$.


Question: The chords $EF$ and $DG$ in the circumcircle of $\triangle DEF$ are parallel because $EF \parallel BC$. This makes $EFGD$ an isosceles trapezoid. Calculate the measure of arc $\widehat{FG}$.


Question: Find the sum of all roots $u$ of the equation $34u^2 - u - 4n = 0$ that lie in the interval $[-1, 1)$, summed over all integers $n$ from 0 to 8.


Question: In the circumcircle of $\triangle DEF$ (Nine-Point Circle of $\triangle ABC$), points $H$ and $J$ are the feet of the altitudes from vertices $B$ and $C$. Calculate the degree measure of the minor arc $\widehat{HJ}$ given $\angle BAC = 84^\circ$.

The problem asks for the degree measure of the minor arc $\widehat{HJ}$ in the circumcircle of $\triangle DEF$, which is


Question: Using $n_1=25$ and $d_1=11$, and the recurrence relations $d_{k+1} = n_k d_k$ and $3 n_{k+1} = n_k^2 - n_k d_k + d_k^2$, compute the values of $n_k$ and $d_k$ modulo 1000 for $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.


Question: In a regular 24-gon, calculate the total number of perfect matchings using chords of equal length by summing the counts for each valid step $k$. There are 4 steps with GCD 1 (count $2^1$ each), 2 steps with GCD 2 (count $2^2$ each), 2 steps with GCD 3 (count $2^3$ each), 1 step with GCD 4 (count $2^4$), 1 step with GCD 6 (count $2^6$), 1 step with GCD 12 (count $1$), and 1 step with GCD 8 (count 0). Compute the sum: $(4 \times 2) + (2 \times 4) + (2 \times 8) + (1 \times 16) + (1 \times 64) + 1$.


Question: The problem asks for the minimum value of f(X) = sum of distances from X to vertices of pentagon ABCDE. Given the geometry, the angle at vertex A is approximately 141.8 degrees, which is greater than 120 degrees. For the geometric median of a polygon, if the internal angle at a vertex is >= 120 degrees, that vertex is the optimal point X. Calculate the sum of distances from A to the other vertices: AB + AC + AD + AE.


Question: Solve the quartic equation $9y^4 - 18y^3 - 143y^2 - 84y + 96 = 0$ to find its real roots, specifically the negative ones, and identify which one fits the form $\frac{a-\sqrt{b}}{c}$ with $a,b,c$ positive integers and $a,c$ coprime.


Question: In triangle ABC with AB=14, BC=7, and angle B=60 degrees, calculate the length of AC. In triangle ADE with AE=26, DE=13, and angle E=60 degrees, calculate the length of AD.


Question: Solve the equation $9y^4 - 18y^3 - 143y^2 - 84y + 96 = 0$ for $y$. Identify the unique negative real root and express it in the form $\frac{a-\sqrt{b}}{c}$ where $a, b, c$ are positive integers and $a, c$ are relatively prime.


Question: In a coordinate system with origin at A, let points on ray AB and ray AC be represented by their distances u, v from A. Points D, E are on AB at u=4, 20 respectively. Points F, G are on AC at v=13, 65 respectively. The quadrilateral DEGF (vertices D-E-G-F in order) has an area of 288. There exists a factor k such that the area of a triangle with vertices A, and two points at distances u, v on the rays is k*u*v. Calculate the value of k.


Question: Solve for $\alpha$ given the equation $28^2 (\cos^2 \alpha + \sin^2(\alpha + 60^\circ)) = 38^2$, restricted to $0 < \alpha < 30^\circ$.


Question: Compute 1 + 27 + 1 + 50/3 + 100.


Question: Categorize the valid step sizes $k \in \{1, \dots, 12\}$ by their GCD with 24 and check validity.


Question: The points $G, D, E, H, J, F$ appear in order on the circumcircle of $\triangle DEF$. The sides $BC, AC, AB$ intersect the circle again at $G, H, J$. Given $BC \parallel EF$, $AC \parallel DF$, and $AB \parallel DE$, determine the arc measures $\widehat{FG}$ and $\widehat{HJ}$ in degrees.


Question: Solve the equation $9y^4 - 18y^3 - 143y^2 - 84y + 96 = 0$ for the unique real root $y$ such that $y < 0$.

To find the real root(s) of the equation $9y^4 - 18y^3 - 143y^2 - 84y + 96 = 0$, we attempt to factor the polynomial.

F


Question: Factor the polynomial $9y^4 - 18y^3 - 143y^2 - 84y + 96$ into two quadratic polynomials with integer coefficients of the form $(3y^2 + ay + b)(3y^2 + cy + d)$.


Question: Given $\triangle ABC$ with angles $84^\circ, 60^\circ, 36^\circ$ and $D, E, F$ as midpoints of $BC, AC, AB$, let $\Gamma$ be the circumcircle of $\triangle DEF$. Let $G, H, J$ be the intersections of $\Gamma$ with $BC, AC, AB$ respectively (other than $D, E, F$). Given that $\widehat{DE} = 72^\circ, \widehat{FG} = 72^\circ, \widehat{HF} = 72^\circ$ on $\Gamma$, find the measure of arc $\widehat{HJ}$ in degrees.


Question: Compute $x_1, x_2, x_3$ as reduced fractions $m_k/n_k$. Calculate $S_k = m_k + n_k \pmod{1000}$ for each and report the exact fraction for $x_1$ and $x_2$. Also, investigate if there is a relationship between $x_k$ and $\cot(2^k \theta)$ or similar trigonometric substitutions.


Question: Find the sum of all positive real numbers $ k $ such that the function $ f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} $ achieves its minimum value at exactly two distinct positive real numbers $ x $. Please show the steps for determining the condition and solving for $ k $.


Question: Is the vertex A the optimal point X for minimizing the sum of distances to the vertices of the pentagon? If so, confirm m=40, n=20, p=3 and output m+n+p.


Question: Calculate $t$, the number of distinct values of $x$ in the interval $0 < x < 2\pi$ that satisfy $\sin(5x) = 1$ or $\sin(5x) = -1$.


Question: In triangle ABC with side lengths AB=14, BC=7 and angle B=60°, calculate the length of diagonal AC. Similarly, in triangle ADE with side lengths AE=26, ED=13 and angle E=60°, calculate the length of diagonal AD.


Question: For one new line segment connecting two random points on different quadrants of a disk divided by 2 perpendicular diameters: what is the expected number of intersection points this segment makes with the 2 diameters?


Question: Calculate the expected number of intersection points between the 25 chords. There are 25 chords, each connecting two points in different quadrants. The number of pairs of chords is $\binom{25}{2}$. Determine the probability of intersection for any pair of chords under these constraints and compute the total expected intersections.


Question: What is the sum of 139 and 10?


Question: In the range $1 \le x \le 3^6$, find the number of triples $(a,b,c)$ such that $3 \nmid a, 3 \nmid b, 3 \nmid c$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Return the answer modulo 1000.


Question: Compute $12^{2^{2024}} \pmod{1000}$.


Question: In triangle $ABC$ with $\angle A = 90^\circ$, let $K$ and $L$ be interior points such that $AK=AL=BK=CL=KL=14$. Find the area of quadrilateral $BKLC$.


Question: In the pentagon with the given parameters, calculate the exact value of f(A) = AB + AC + AD + AE and confirm it is 40 + 20\sqrt{3}. Also check the value of f(C) = AC + BC + CD + CE + CE (wait, f(C) includes CX=0).


Question: Calculate the sum of the $y$-coordinates for the second branch intersections. Consider the equation $34u^2 - 137u + 136 - 4n = 0$ for integers $n \ge 0$. A solution $u$ is valid if $u \in [1, 3)$ and $x=4n+u \ge 0$. The $y$-coordinate corresponding to each valid $u$ is $2-u$. Determine the valid values of $n$, identify all valid roots $u$, calculate their corresponding $y$-coordinates, and return the total sum of these $y$-coordinates.


Question: In the circumcircle of the medial triangle DEF (which is the Nine-Point Circle), with triangle ABC having angles 84, 60, and 36 degrees, what are the degree measures of the minor arcs $\widehat{HJ}$ (connecting altitude feet on AB and AC) and $\widehat{FG}$ (connecting midpoint F on AB and altitude foot G on BC)?


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


Question: Calculate the number of distinct real values $x$ in the interval $0 < x < 2\pi$ such that the function $f(x) = \sin(7\pi \sin(5x))$ satisfies $f(x) = 0$ and $f'(x) = 0$. Provide this count as the value $t$.


Question: Find the number of distinct real values of $x$ in the open interval $(0, 2\pi)$ that satisfy the equation $\sin(7\pi \sin(5x)) = 0$. Let this count be $n$.


Question: In a regular 24-gon, find the total number of ways to draw 12 segments of equal length (step k) such that each vertex is an endpoint of exactly one segment. Compute this number for each step $k \in \{1, \dots, 12\}$ by calculating the number of valid perfect matchings on the induced cycle graph for that step, and return the sum of these counts.


Question: Given $k = \frac{3}{13}$ is the area scaling factor such that $\text{Area}(\triangle APQ) = k \cdot AP \cdot AQ$ for points $P, Q$ on rays $AB$ and $AC$ respectively. Points on ray $AB$: $D(4), E(20), B(28)$. Points on ray $AC$: $F(13), G(65), C(91)$. $M$ is reflection of $D$ through $F$. $N$ is reflection of $G$ through $E$. Vertices of Heptagon are ordered $A, F, N, B, C, E, M$. Calculate the area of this Heptagon.


Question: Calculate the area of the quadrilateral BKLC defined by: Right triangle ABC ($\angle A=90^\circ, BC=38$), $AK=AL=14$, $BK=14$, $CL=14$, $KL=14$. Express the area as $n\sqrt{3}$ and report the integer $n$.


Question: Determine the value of $n$ by solving the geometric constraints. Specifically, calculate $n$ derived from Area($BKLC$) = Area($ABC$) - Area($ABK$) - Area($ACL$) - Area($AKL$).


Question: Determine the coefficients of the polynomial $Q(x)$ defined by $x \frac{d}{dx}\left((x-18)(x-72)(x-98)(x-k)\right) - (x-18)(x-72)(x-98)(x-k) = 0$. Specifically, express $Q(x) = A x^4 + B x^3 + C x^2 + D x + E$ with coefficients $A, B, C, D, E$ explicitly in terms of $k$. Use $S_1 = \sum r_i$, $S_2 = \sum r_i r_j$, $S_4 = \prod r_i$.


Question: (Executor result needed)


Question: In a 3x9 grid filled with numbers 1-9 where each row contains all numbers 1-9 once and each 3x3 block (consisting of 3 consecutive columns) also contains all numbers 1-9 once: How many distinct values can appear in the first block's cells (row 0, cols 0-2)?


Question: Calculate the total number of ways by summing the number of perfect matchings for each valid chord length $k \in \{1, \dots, 12\}$. Specifically, determine the number of ways for each $k$ and sum them.


Question: What is the probability that two chords in a disk, where each chord connects two points chosen uniformly at random on the perimeter such that the points belong to different quadrants, intersect at a point strictly inside the disk?


Question: Find the polynomial equation in k that characterizes the condition that f(x) achieves its minimum at exactly two points.


Question: I need to confirm the final expression form $\frac{a+b\sqrt{c}}{d}$ with $a,b,c,d$ positive integers and $gcd(a,b,d)=1$.

\text{The form is confirmed as the unique reduced representation.}


Question: Calculate the number of ways to draw 12 segments of equal lengths covering all 24 vertices of a regular 24-gon, where each valid configuration consists of edges of a single chord length $k$ (for $k \in \{1, \dots, 12\}$). Sum the number of perfect matchings for each distinct geometric length $k$ to provide the final answer.


Question: Calculate the area of the quadrilateral $BKLC$ for a right triangle $\triangle ABC$ ($\angle A=90^\circ, BC=38$) containing points $K,L$ such that $AK=AL=BK=CL=KL=14$. The area is expressed as $n\sqrt{3}$. Compute the numeric value of the area first, then deduce $n$.


Question: Please compute the sequence of rational numbers $x_k$ starting with $x_1 = 25/11$ and iterating $x_{k+1} = \frac{1}{3} \left( x_k + \frac{1}{x_k} - 1 \right)$ up to $k=2025$. Express $x_{2025}$ as an irreducible fraction $m/n$ with positive integers $m, n$. Then compute $(m + n) \pmod{1000}$ and return the result.


Question: Calculate the exact sum of all $y$-coordinates of the intersection points between the parabola $x=34y^2$ and the periodic function $f(x)$ with period 4. Provide the simplified sum in the form $\frac{P}{Q}$ or $\frac{A+B\sqrt{C}}{D}$ where $A,B,C,D$ are rational numbers, and then determine the integer form $a,b,c,d$ with $a,b,c,d$ positive and $\gcd(a,b,d)=1$.


Question: We established the matching count formula. Calculate the number of matchings for $k=12$ (where chord length is diameter, cycle length $L=2$).


Question: Calculate the probability $p$ that a line segment connecting two points chosen uniformly at random from the perimeter of a disk (with the constraint that the two points lie in different quadrants defined by two perpendicular diameters) intersects a second line segment connecting two other points chosen uniformly at random from the perimeter (also satisfying the quadrant constraint).


Question: Find the point X that minimizes the sum of distances $AX+BX+CX+DX+EX$ for the vertices of the described pentagon. Express the minimum value as $m+n\sqrt{p}$.


Question: Calculate the sum of the roots of the polynomial equation in $k$ derived from setting the discriminant of $Q(x) = 3x^4 + (-376-2k)x^3 + (10116+188k)x^2 - 127008k$ to zero.


Question: For the polynomial $Q(x) = 3x^4 + (-376-2k)x^3 + (10116+188k)x^2 - 127008k$, determine the values of $k$ such that $Q(x)$ has a multiple root (i.e., $\gcd(Q(x), Q'(x)) \neq 1$). Specifically, find the polynomial in $k$ obtained by eliminating $x$ from the system $Q(x)=0$ and $Q'(x)=0$.


Question: Calculate the sum of the values of $k$ for which the polynomial $Q(x) = 3x^4 + (-376-2k)x^3 + (10116+188k)x^2 - 127008k$ has a multiple root. Specifically, find the equation in $k$ satisfied by these values and determine the sum of its roots.


Question: Given that $x_1, x_2$ are the distinct roots of the quadratic equation $3x^2 - 2(188+k)x + (10116+188k) = 0$, and $f(x) = \frac{(x-18)(x-72)(x-98)(x-k)}{x}$, derive the explicit polynomial equation in terms of $k$ that results from the condition $f(x_1) = f(x_2)$. (Do not solve for $k$, just find the polynomial).


Question: In $\triangle ABC$ with $\angle A = 84^\circ, \angle B = 60^\circ, \angle C = 36^\circ$, let $D, E, F$ be midpoints of $BC, AC, AB$. Let $\Gamma$ be the circumcircle of $\triangle DEF$. $G \in BD \cap \Gamma$, $H \in AE \cap \Gamma$, $J \in AF \cap \Gamma$. What are the measures of minor arcs $\widehat{DE}$, $\widehat{FG}$, and $\widehat{HJ}$?


Question: Given $x_1, x_2$ are roots of the quadratic $3x^2 - 2(188+k)x + (10116+188k) = 0$ and $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$, find the equation satisfied by $k$ imposed by the condition $f(x_1) = f(x_2)$.


Question: Given $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$, let $x_1, x_2$ be the roots of $3x^2 - 2(188+k)x + (10116+188k) = 0$. Assume $x_1, x_2$ are the x-coordinates of the global minimum. What is the condition on $k$ derived from $f(x_1) = f(x_2)$?


Question: Calculate the sum of the three positive real values of $k$ such that the function $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$ attains its minimum value at exactly two distinct points.


Question: Given $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$, substitute the roots of the equation $3x^2 - 2(188+k)x + (10116+188k) = 0$ into $f(x)$ to obtain the values of the function at the local minima. Simplify the condition $f(x_1) = f(x_2)$ to find the exact polynomial equation in $k$.


Question: Let $x_1, x_2$ be the roots of $3x^2 - 2(188+k)x + (10116+188k) = 0$. Given $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$, find the polynomial equation satisfied by $k$ under the condition $f(x_1) = f(x_2)$.


Question: In the equation $3x^2 - 2(188+k)x + (10116+188k) = 0$, let $x_1$ and $x_2$ be the two positive real roots. Given the function $f(x) = \frac{(x-18)(x-72)(x-98)(x-k)}{x}$, find the algebraic equation in $k$ that results from the condition $f(x_1) = f(x_2)$.


Question: Solve the equation formed by the condition that the local minima in the two negative intervals of $f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$ have equal values. Express this condition as a polynomial equation in $k$.


Question: Calculate the exact value of the sum $\sum_{m=1}^8 \sqrt{1+544m}$.


Question: Solve the system of equations for $y$-coordinates where $x=34y^2$ and $y=f(x)$ holds for $x \in [0, 34)$, given $f(x)$ is periodic with period 4, $f(x)=x$ for $x \in [-1, 1)+4k$, and $f(x)=2-x$ for $x \in [1, 3)+4k$. Find the total sum of these $y$ values.


Question: Calculate the exact value of the summation $\sum_{m=1}^8 \sqrt{1+544m}$ and simplify the resulting expression $S = \frac{10}{68} + \frac{1}{68}\sum_{m=1}^8 \sqrt{1+544m}$ into the form $\frac{a+b\sqrt{c}}{d}$ satisfying the conditions ($a,b,d$ coprime, $c$ square-free). Report the value $a+b+c+d$.


Question: Compute the value of the sum $S = \frac{10}{68} + \frac{1}{68}\sum_{m=1}^8 \sqrt{1+544m}$ and simplify it into the form $\frac{a+b\sqrt{c}}{d}$ as described in the problem statement.