sungyub/dapo-math-17k-verl huggingface dataset:
user: 若 $\triangle A B C$ 的三边长分别是 $2,3,4$, 其外心为 $O$, 求 $\overrightarrow{O A} \cdot \overrightarrow{A B}+\overrightarrow{O B} \cdot \overrightarrow{B C}+$ $\overrightarrow{O C} \cdot \overrightarrow{C A}$ 的值。原始答案的形式为\frac{k}{m}，请给出k+m的值。

Answer: -27
user: 设 $\triangle A B C$ 的三边长分别为 $B C=a, C A=b, A B=c$. 若 $c=2, C=\frac{\pi}{3}$, 且 $\sin C+\sin (B-A)-2 \sin 2 A=0$, 求 $\triangle A B C$ 的周长。如果答案是 $x + y\sqrt{z}$，请给出 $x + y + z$ 的值。

Answer: 7
user: 在长方体 \(A B C D-A_{1} B_{1} C_{1} D_{1}\) 中，\(A B=2, A D=A A_{1}=1\)，求点 \(A\) 到平面 \(A_{1} B D\) 的距离。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

Answer: 5
user: 从正15边形的顶点中选出3个连成钝角三角形，不同的选法有多少种？

Answer: 315
user: 若 \(x^{2}+y^{2} \leqslant 1\)，求 \(x^{2}+x y-y^{2}\) 的最大值和最小值，并将结果表示为一个区间 \([- \frac{\sqrt{m}}{n}, \frac{\sqrt{m}}{n}]\)，其中 \(m\) 和 \(n\) 为正整数。请给出 m 和 n 的和。

Answer: 7
user: 设复数 $z_{1}, z_{2}$ 在复平面内对应的点分别为 $Z_{1}, Z_{2}, O$ 为坐标原点, 若 $\left|z_{1}\right|=1$, $5 z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}=0$, 求 $\triangle O Z_{1} Z_{2}$ 的面积。

Answer: 1
user: 求值: \(\lim _{n \rightarrow+\infty}\left(\sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\right)=(\quad)\pi\)。如果答案是\(\frac{k}{m}\pi\)，请填写\(k+m\)的值。

Answer: 7
user: 设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leqslant x_{i} \leqslant 1(1,2, \cdots, 21)$, 求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。

Answer: 220
user: 在 $\triangle A B C$ 中, $A=90^{\circ}, A B=1, A C=\sqrt{3}$, 点 $P$ 满足 $\frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。如果答案是 $x^\circ$，求 $x$ 的值。

Answer: 120
user: 设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leq x_{i} \leq 1(1,2, \cdots, 21)$ ，求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。

Answer: 220
user: 设复数 $z_{1}, z_{2}$ 在复平面内对应的点分别为 $Z_{1}, Z_{2}$，$O$ 为坐标原点，若 $\left|z_{1}\right|=1,5 z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}=0$，则 $\triangle O Z_{1} Z_{2}$ 的面积为多少？

Answer: 1
user: 设复数 $z$ 满足 $|3 z-7 \mathrm{i}|=3$, 求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为 $\frac{k}{m}$，请给出 $k + m$ 的值。

Answer: 11
user: 已知数列 \(\left\{a_{n}\right\}\) 的前 \(n\) 项和 \(S_{n}=(-1)^{n} \cdot a_{n}+\frac{1}{2^{n}}+n-3\)，且实数 \(t\) 满足 \(\left(t-a_{n}\right)\left(t-a_{n+1}\right)<0\)，求 \(t\) 的取值范围。如果 \(t\) 的取值范围是 \((x, y)\)，请将其写成 \(x = -\frac{k}{m}\) 和 \(y = \frac{n}{p}\) 的形式，并求出 \(k+m+n+p\) 的值。

Answer: 22
user: 设 $A, B$ 分别是 $x$ 轴, $y$ 轴上的动点, 若以 $A B$ 为直径的圆 $C$ 与直线 $2 x+y-4=0$ 相切, 则圆 $C$ 面积的最小值为 $\frac{k}{m}\pi$，请计算k+m的值。

Answer: 9
user: 设多项式 \(f(x)\) 的各项系数都是非负实数，且 \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=1\)。求 \(f(x)\) 的常数项的最小值。答案的格式为\(\frac{k}{m}\)，请给出k+m的值。

Answer: 4
user: 设 \(\alpha, \beta\) 为锐角, 且 \(\cos (\alpha+\beta)=\frac{\sin \alpha}{\sin \beta}\), 求 \(\tan \alpha\) 的最大值。答案应为\(\frac{\sqrt{k}}{m}\)的形式，请给出k+m的值。

Answer: 6
user: 已知 $A, B, C$ 是集合 $\{1,2, \cdots, 2020\}$ 的子集，且满足 $A \subseteq B \subseteq C$ ，求这样的有序组 $(A, B, C)$ 的总数。答案的形式为 $x^y$，请给出x和y的和。

Answer: 2024
user: 设随机变量 \(X\) 的概率分布列为 \(P(x=k)=\frac{1}{2^{k}}(k=1,2,3, \cdots)\)，\(Y\) 表示 \(X\) 被 3 除的余数，求随机变量 \(Y\) 的数学期望 \(E Y\)。答案的形式为 \(\frac{k}{m}\)，请给出k+m的值。

Answer: 15
user: 已知 $M=\left\{(x, y) \mid y \geqslant x^{2}\right\}, N=\left\{(x, y) \mid x^{2}+(y-a)^{2} \leqslant 1\right\}$. 求使 $M \cap N=N$ 成立的 $a$ 的最小值。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

Answer: 9
user: A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. Find the smallest value that the range (largest observation minus smallest) can assume for such a sample.

Answer: 5
user: If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then find the value of \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor.\]

Answer: 8204
user: Square \(ABCD\) has side length \(2\). A semicircle with diameter \(\overline{AB}\) is constructed inside the square, and the tangent to the semicircle from \(C\) intersects side \(\overline{AD}\) at \(E\). The length of \(\overline{CE}\) can be expressed in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please find the value of \(k + m\).

Answer: 7
user: Let \(a\) and \(b\) be real numbers such that \(\sin a + \sin b = \frac{\sqrt{2}}{2}\) and \(\cos a + \cos b = \frac{\sqrt{6}}{2}\). Find the value of \(\sin(a+b)\). The original answer is in the form \(\frac{k\sqrt{n}}{m}\), please give the value of \(k + m + n\).

Answer: 6
user: In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie, and no points for a loss, find the number of games that ended in a tie.

Answer: 5
user: The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. Find the length of the shortest side of the triangle.

Answer: 17
user: There exist positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that:
\[A \log_{200} 5 + B \log_{200} 2 = C. \]
Find the value of $A + B + C$.

Answer: 6
user: Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie, Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

Answer: 2
user: In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result.

Answer: 625
user: Find the number of real solutions to the equation \( \frac{x}{100} = \sin x \).

Answer: 63
user: The six-digit number $20210A$ is prime for only one digit $A$. What is the value of $A$?

Answer: 9
user: How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 15\}$ have the following two properties?

No two consecutive integers belong to $S$.
If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

Find the number of such subsets.

Answer: 405
user: Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. Find the probability that no two adjacent people will stand. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 303
user: Six distinct integers are picked at random from \(\{1,2,3,\ldots,10\}\). What is the probability that, among those selected, the second smallest is \(3\)? Express your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest form, and find the value of \(k + m\).

Answer: 4
user: Triangle \( ABC \) has side lengths in arithmetic progression, and the smallest side has length \( 6 \). If the triangle has an angle of \( 120^\circ \), what is the area of \( ABC \)? The original answer is in the form of \( k\sqrt{m} \). Please provide the value of \( k + m \).

Answer: 18
user: Let the sequence \((a_n)\) be defined as \(a_1=\frac{1}{3}\) and \(a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}\) for every \(n\geq 1\). If \(a_k\) is the largest term of the sequence satisfying \(a_k < \frac{1}{50}\), find the value of \(k\).

Answer: 193
user: What is the number of terms with rational coefficients among the $1001$ terms of the expression $(x \sqrt[3]{2} + y \sqrt{3})^{1000}$?

Answer: 167
user: $ABCD$ is a square with a side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ such that $AE = AF$, and the quadrilateral $CDFE$ has maximum area. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 13
user: In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle, and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees? If the answer is $x^\circ$, what is the value of $x$?

Answer: 15
user: For any complex number \(w = a + bi\), \(|w|\) is defined to be the real number \(\sqrt{a^2 + b^2}\). If \(w = \cos{40^\circ} + i\sin{40^\circ}\), find the value of 
\[ x = |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \]
in the form \(\frac{k}{l}\sin{\phi^\circ}\), where \(k\), \(l\) are integers, and \(\phi\) is an angle in degrees. Please give the value of \(k + l + \phi\).

Answer: 31
user: In the figure, $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AD}$, and $F$ is on $\overline{BE}$. If $\overline{CF}$ is perpendicular to $\overline{BE}$, find the area of quadrilateral $CDEF$. The original answer is in \frac{k}{m} format, where k and m are coprime integers. Please give the value of k + m.

Answer: 16
user: The cost $C$ of sending a parcel post package weighing $P$ pounds, where $P$ is an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. What is the formula for the cost $C$ in terms of $P$? The original answer is in the form $C = a + b(P - 1)$, please give the value of $a + b$.

Answer: 13
user: Find the number of distinct solutions of the equation $\big|x-|2x+1|\big| = 3$.

Answer: 2
user: Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, find the value of $n$.

Answer: 9
user: A circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of the circular piece. Find the total amount of metal wasted as a fraction of the area of the original square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 3
user: Let \((1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\) be an identity in \(x\). If we let \(s = a_0 + a_2 + a_4 + \ldots + a_{2n}\), the original answer is in the form \(\frac{k \cdot 3^n + m}{n}\), where n cannot be divisible by 2. Please find the value of \(k + m + n\).

Answer: 4
user: Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?

Answer: 15
user: Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? If the answer is \(\sqrt{x}\), what is the value of \(x\)?

Answer: 95
user: 定义序列 $\{a_n\}_{n\geq 1}$ 如下：
- 当 $n \leq 2$ 时，$a_n = n - 1$
- 对于所有 $n \geq 2$，$a_n$ 是 $a_{n-1} + a_{n-2}$ 除以 $3$ 的余数

计算从 $a_{2018}$ 到 $a_{2025}$ 的项的和 $\sum_{i=2018}^{2025} a_i$。

Answer: 9
user: Let $P(x) = 1 - x + x^2 - x^3 + \dots + x^{18} - x^{19}$ and $Q(x) = P(x - 1)$. Find the coefficient of $x^2$ in polynomial $Q$.

Answer: 1140
user: How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions?

(i) $4000 \leq N < 6000;$ 
(ii) $N$ is a multiple of 5; 
(iii) $3 \leq b < c \leq 6$.

Find the number of such numbers.

Answer: 24
user: Let $ABCD$ be a quadrilateral with sides $|\overline{AB}|=2$, $|\overline{BC}|=|\overline{CD}|=4$, and $|\overline{DA}|=5$. The opposite angles, $\angle A$ and $\angle C$, are equal. What is the length of diagonal $BD$? The original answer is in $k\sqrt{m}$ format, please give the value of k + m.

Answer: 8
user: How many ordered pairs of integers $(x, y)$ satisfy the equation $$x^{2020} + y^2 = 2y?$$

Answer: 4
user: Square $BDEC$ with center $F$ is constructed on the outside of triangle $ABC$ such that $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$. If $[AF] \cap [BC] = \{G\}$, find the length of $|BG|$. The original answer is in the form $m - n\sqrt{p}$, please give the value of $m + n + p$.

Answer: 11
user: How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters?

Answer: 60
user: What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$? Find the largest possible area.

Answer: 27
user: Ali and Veli go hunting. The probability that each will successfully hit a duck is \(\frac{1}{2}\) on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. What is the probability that Veli hits more ducks than Ali? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form, and find the value of k + m.

Answer: 3
user: We call a number $10^3 < n < 10^6$ a balanced number if the sum of its last three digits is equal to the sum of its other digits. Find the sum of all balanced numbers in $\bmod{13}$.

Answer: 0
user: How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$? Provide your answer as an integer.

Answer: 60
user: A six-digit number (base 10) is considered "squarish" if it satisfies the following conditions:

None of its digits are zero.
It is a perfect square.
The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers.

How many squarish numbers are there? Provide your answer as an integer.

Answer: 2
user: Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. Find the area of the region in the plane covered by these squares.

Answer: 175
user: A pyramid of non-negative integers is constructed as follows:

The first row consists of only $0$.
The second row consists of $1$ and $1$.
The $n^{th}$ row (for $n > 2$) is an array of $n$ integers, where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row.

Let $S_n$ be the sum of numbers in the $n^{th}$ row. Determine the remainder when dividing $S_{2018}$ by $2018$.

Answer: 2
user: Find the number of 3-digit numbers not containing the digit 0 and such that one of the digits is the sum of the two others.

Answer: 108
user: How many integer pairs $(m,n)$ are there satisfying the equation $4mn(m+n-1)=(m^2+1)(n^2+1)$? Provide your answer as an integer.

Answer: 5
user: A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two-hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. At what time will the privateer overtake the merchantman? Provide your answer in the format of H:MM PM, where H and MM are two-digit numbers (possibly with leading zeros). Please give the value of H + MM.

Answer: 35
user: Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. What is the area of the region of the plane that is covered by the union of the five triangular regions? Express your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of a + b.

Answer: 15
user: Each of 100 students sends messages to 50 different students. Find the least number of pairs of students who send messages to each other.

Answer: 50
user: Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 7
user: The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?

Answer: 12
user: An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes $5$ seconds for the first car to pass the observer. If the time it takes for the $10^{\text{th}}$ car to pass is $x$ seconds, find the value of $100x$.

Answer: 81
user: When simplified, \( (x^{-1} + y^{-1})^{-1} \) is equal to what expression in the form \(\frac{kx}{my}\)? Please find the value of k + m.

Answer: 2
user: Square $PQRS$ lies in the first quadrant. Points $(3,0)$, $(5,0)$, $(7,0)$, and $(13,0)$ lie on lines $SP$, $RQ$, $PQ$, and $SR$, respectively. If the sum of the coordinates of the center of the square $PQRS$ is $x$, what is the value of $10x$?

Answer: 64
user: What is the $33$-rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$?

Answer: 0
user: A box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that the sum of the numbers on the balls drawn is odd. Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and provide the value of k + m.

Answer: 349
user: If two poles $20''$ and $80''$ high are $100''$ apart, find the height (in inches) of the intersection of the lines joining the top of each pole to the foot of the opposite pole.

Answer: 16
user: What is the largest volume of a sphere which touches a unit sphere internally and touches externally a regular tetrahedron whose corners are over the unit sphere? Express your answer as a fraction in the form \(\frac{k}{m}\), and give the value of \(k + m\).

Answer: 4
user: Corners are sliced off a unit cube so that the six faces each become regular octagons. The original answer for the total volume of the removed tetrahedra is in the form \(\frac{k - m\sqrt{2}}{n}\). Please find the value of k + m + n.

Answer: 20
user: What is the radius of a circle inscribed in a rhombus with diagonals of length \(10\) and \(24\)? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please find the value of \(k + m\).

Answer: 73
user: Let $f(x) = \sqrt{\sin^4 x + 4\cos^2 x} - \sqrt{\cos^4 x + 4\sin^2 x}$. Find an equivalent form of $f(x)$ in the format $k\cos mx$. Please provide the value of $k + m$.

Answer: 3
user: How many integer pairs \((x, y)\) are there such that:
\[0 \leq x < 165, \quad 0 \leq y < 165 \text{ and } y^2 \equiv x^3 + x \pmod{165}?\]
Find the number of such integer pairs.

Answer: 99
user: In a class, some pupils learn German, while others learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. Besides the girls learning French, there are 16 pupils. How many pupils are there in the class?

Answer: 23
user: Let $A$, $B$ be the number of digits of $2^{1998}$ and $5^{1998}$ in the decimal system. Find the value of $A + B$.

Answer: 1999
user: A digital watch displays hours and minutes with $\text{AM}$ and $\text{PM}$. What is the largest possible sum of the digits in the display?

Answer: 23
user: What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white, and $2$ blue balls? Provide your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Calculate the value of \(k + m\).

Answer: 31
user: Find the number of solution-pairs in the positive integers of the equation $3x + 5y = 501$.

Answer: 33
user: In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? Provide your answer in the form of a fraction \frac{k}{m}, and then calculate the value of k + m.

Answer: 7
user: In $\triangle ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Circle $P$ is the circle with the smallest radius that passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$, of circle $P$ with sides $AC$ and $BC$, respectively. If the length of segment $QR$ is $x$, what is the value of $10x$?

Answer: 48
user: Albert, Bob, and Charlie are each thinking of a number. Albert's number is one more than twice Bob's. Bob's number is one more than twice Charlie's, and Charlie's number is two more than twice Albert's. What number is Albert thinking of? The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: -4
user: Pikachu, Charmander, and Vulpix are three of the four equally-skilled players in a Pokemon bracket tournament. Because they are equally skilled, whenever any two of the players battle, they are equally likely to win. In the bracket tournament, the four players are randomly paired into two rounds, each round consisting of two players. The winners of the first two rounds then play each other in the final round. The winner of the final match ranks first; the loser of the final round ranks second; and the two losers of the previous rounds jointly rank third. What is the probability that Charmander plays Vulpix in a round, but ranks lower than Pikachu? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and provide the value of \(k + m\).

Answer: 9
user: Vessel $A$ contains liquids $X$ and $Y$ in the ratio $X:Y = 8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y = 5:9$. In what ratio should you mix the liquids from both vessels to achieve a mixture with the ratio $X:Y = 1:1$? Express your answer in the form of $a:b$, and provide the value of $a + b$.

Answer: 37
user: Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \sqrt{2} \)? Find the value of \( x \).

Answer: 56
user: Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. If the number of seconds he has to spare when he gets to the tunnel entrance is $x$, find the value of $10x$.

Answer: 144
user: Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$.

Answer: 3
user: Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?

Answer: 17
user: Find the sum of all positive integers $n$ such that \(\frac{n^3+8n^2+8n+80}{n+7}\) is an integer.

Answer: 66
user: The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?

\[
\begin{array}{cccc}
& 6 & 4 & 1 \\
& 8 & 5 & 2 \\
+ & 9 & 7 & 3 \\
\hline
2 & 4 & 5 & 6
\end{array}
\]

Provide the largest digit that can be changed.

Answer: 7
user: Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once?

Answer: 16