prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$

answer: 70


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$.

answer: 588


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$

answer: 16


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.

answer: 117


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
There are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, 7, 8$ exactly once. Let $N$ be the number of these integers that are divisible by $22$. Find the difference between $N$ and $2025$.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
There are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, 7, 8$ exactly once. Let $N$ be the number of these integers that are divisible by $22$. Find the difference between $N$ and $2025$.$

answer: 279


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$

answer: 504


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

answer: 821


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $k$ be a real number such that the system \begin{align*} &|25 + 20i - z| = 5 \ &|z - 4 - k| = |z - 3i - k| \end{align*} has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Here $i = \sqrt{-1}$.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $k$ be a real number such that the system \begin{align*} &|25 + 20i - z| = 5 \ &|z - 4 - k| = |z - 3i - k| \end{align*} has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Here $i = \sqrt{-1}$.$

answer: 77


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.

answer: 62


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle. [asy] unitsize(20);  add(grid(9,3));  draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));  real a = 0.5;  label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a));  label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a));  label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a));  [/asy] The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle. [asy] unitsize(20);  add(grid(9,3));  draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));  real a = 0.5;  label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a));  label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a));  label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a));  [/asy] The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.

answer: 81


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of all these intersection points can be expressed in the form $\tfrac{a + b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$, $b$, $d$ have greatest common divisor equal to $1$, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. Graph [asy] import graph;  size(300);  Label f;  f.p=fontsize(6);  yaxis(-2,2,Ticks(f, 2.0));  xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((0, 0)..(1/4,sqrt(1/136))..(1/2,sqrt(1/68))..(0.75,sqrt(0.75/34))..(1, sqrt(1/34))..(2, sqrt(2/34))..(3, sqrt(3/34))..(4, sqrt(4/34))..(5, sqrt(5/34))..(6, sqrt(6/34))..(7, sqrt(7/34))..(8, sqrt(8/34)), red); draw((0, 0)..(1/4,-sqrt(1/136))..(0.5,-sqrt(1/68))..(0.75,-sqrt(0.75/34))..(1, -sqrt(1/34))..(2, -sqrt(2/34))..(3, -sqrt(3/34))..(4, -sqrt(4/34))..(5, -sqrt(5/34))..(6, -sqrt(6/34))..(7, -sqrt(7/34))..(8, -sqrt(8/34)), red); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp);  draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black);  draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of all these intersection points can be expressed in the form $\tfrac{a + b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$, $b$, $d$ have greatest common divisor equal to $1$, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. Graph [asy] import graph;  size(300);  Label f;  f.p=fontsize(6);  yaxis(-2,2,Ticks(f, 2.0));  xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((0, 0)..(1/4,sqrt(1/136))..(1/2,sqrt(1/68))..(0.75,sqrt(0.75/34))..(1, sqrt(1/34))..(2, sqrt(2/34))..(3, sqrt(3/34))..(4, sqrt(4/34))..(5, sqrt(5/34))..(6, sqrt(6/34))..(7, sqrt(7/34))..(8, sqrt(8/34)), red); draw((0, 0)..(1/4,-sqrt(1/136))..(0.5,-sqrt(1/68))..(0.75,-sqrt(0.75/34))..(1, -sqrt(1/34))..(2, -sqrt(2/34))..(3, -sqrt(3/34))..(4, -sqrt(4/34))..(5, -sqrt(5/34))..(6, -sqrt(6/34))..(7, -sqrt(7/34))..(8, -sqrt(8/34)), red); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp);  draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black);  draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]

answer: 259


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities\[x-yz<y-zx<z-xy\]forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities\[x-yz<y-zx<z-xy\]forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

answer: 510


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.

answer: 204


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

answer: 60


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

answer: 735


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$

answer: 468


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$.

answer: 49


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy]         size(4cm);     defaultpen(linewidth(1.2));     draw((0, 0) -- (2, 0) -- (2, 1));     draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2));     draw((0, 0) -- (0, 1), dotted);     draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted);     draw((0, 1) -- (0, 2) -- (1, 2), dotted); [/asy]

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy]         size(4cm);     defaultpen(linewidth(1.2));     draw((0, 0) -- (2, 0) -- (2, 1));     draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2));     draw((0, 0) -- (0, 1), dotted);     draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted);     draw((0, 1) -- (0, 2) -- (1, 2), dotted); [/asy]

answer: 82


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

answer: 106


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $	riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees.[asy] import olympiad; size(6cm); defaultpen(fontsize(10pt)); pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B); guide circ = circumcircle(D, E, F); pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0]; draw(B--A--C--cycle); draw(D--E--F--cycle); draw(circ);  dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);dot(G);dot(H);dot(J); label("$A$", A, (0, .8)); label("$B$", B, (-.8, -.8)); label("$C$", C, (.8, -.8)); label("$D$", D, (0, -.8)); label("$E$", E, (.8, .2)); label("$F$", F, (-.8, .2)); label("$G$", G, (0, .8)); label("$H$", H, (-.2, -1));label("$J$", J, (.2, -.8)); [/asy]

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $	riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees.[asy] import olympiad; size(6cm); defaultpen(fontsize(10pt)); pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B); guide circ = circumcircle(D, E, F); pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0]; draw(B--A--C--cycle); draw(D--E--F--cycle); draw(circ);  dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);dot(G);dot(H);dot(J); label("$A$", A, (0, .8)); label("$B$", B, (-.8, -.8)); label("$C$", C, (.8, -.8)); label("$D$", D, (0, -.8)); label("$E$", E, (.8, .2)); label("$F$", F, (-.8, .2)); label("$G$", G, (0, .8)); label("$H$", H, (-.2, -1));label("$J$", J, (.2, -.8)); [/asy]

answer: 336


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle {DGF}$ and $\triangle {CHG}$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray);  draw(B--C); draw(A--D);  draw(E--F--G--H--cycle);         draw(circle((0,0), 15)); draw(circle(A, 6)); dot(A);  dot(B); dot(C); dot(D);dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8));  label("$E$", E, (.8, -.8));  label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle {DGF}$ and $\triangle {CHG}$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray);  draw(B--C); draw(A--D);  draw(E--F--G--H--cycle);         draw(circle((0,0), 15)); draw(circle(A, 6)); dot(A);  dot(B); dot(C); dot(D);dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8));  label("$E$", E, (.8, -.8));  label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]

answer: 293


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

answer: 237


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.

answer: 610


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.

answer: 149


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.

answer: 907


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

answer: 113


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.

answer: 19


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.

answer: 248


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

answer: 104


prompt_standard:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

Problem:
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.

prompt_implicit:
Solve the problem step by step. End your solution with the final answer as \boxed{<final_answer>}.

In addition to your normal reasoning, you may format your reasoning with implicit thought tags of the following format:

 <implicit_thought>number</implicit_thought>

Where number is a number between 1 and 5, and corresponds to the complexity of the thought. When you use this tool, you can skip the sentence that you would have said and move on with your reasoning. This will allow you to think a lot more about the story, while skipping over obvious conclusions/reasoning steps. We recommend doing this in the center of your reasoning, and not at the beginning or end.

Problem:
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.

answer: 240