sungyub/dapo-math-17k-verl huggingface dataset:
user: A quadratic polynomial with real coefficients and leading coefficient \(1\) is called \(\emph{disrespectful}\) if the equation \(p(p(x))=0\) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \(\tilde{p}(x)\) for which the sum of the roots is maximized. The original answer for \(\tilde{p}(1)\) is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 21
user: What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, \ldots, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)

Find the average number of pairs of consecutive integers in such a subset. The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).

Answer: 5
user: Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy]

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m+n+p$.

Answer: 81
user: Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is the length of $CF$?

Answer: 30
user: Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find the value of $a+b.$

Answer: 85
user: Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N$?

Answer: 9
user: Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. Find the value of $c_1$.

Answer: 11
user: For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$

Answer: 2
user: What is the value of 
$$3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
The answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 142
user: What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).

Answer: 142
user: In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. Find the area of $ABCD$. (Note: The figure is not drawn to scale.)

Answer: 20
user: How many three-digit positive integers have an odd number of even digits? Provide your answer as an integer.

Answer: 450
user: When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?

Answer: 4
user: A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. Find the number of seconds after $4:58$ when the donkey’s $700$th hiccup occurs.

Answer: 15
user: The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?

Answer: 6
user: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? Provide the number of such values.

Answer: 8
user: Square \(ABCD\) has side length \(1\). Points \(P\), \(Q\), \(R\), and \(S\) each lie on a side of \(ABCD\) such that \(APQCRS\) is an equilateral convex hexagon with side length \(s\). The original answer is in the form \(a - \sqrt{b}\). Please give the value of \(a + b\).

Answer: 4
user: The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\]
Find the number of points $P$ with integer coordinates such that the taxicab distance between $P$ and the origin is less than or equal to $20$.

Answer: 841
user: Consider the following $100$ sets of $10$ elements each:
\begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}
Find how many of these sets contain exactly two multiples of $7$.

Answer: 42
user: What is the value of \(\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?\) The answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).

Answer: 232
user: On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. 
 
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes. 
 
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes. 
 
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes. 
 
How many pieces of candy in all did the principal give to the children who always tell the truth?

Answer: 7
user: The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that $$2^{a_7}=2^{27} \cdot a_7.$$ Find the minimum possible value of $a_2$.

Answer: 12
user: Regular hexagon \(ABCDEF\) has side length \(2\). Let \(G\) be the midpoint of \(\overline{AB}\), and let \(H\) be the midpoint of \(\overline{DE}\). Find the perimeter of quadrilateral \(GCHF\). The original answer is in the form \(k\sqrt{m}\). Please provide the value of \(k + m\).

Answer: 11
user: Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with?

Answer: 44
user: On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. 
 
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes. 
 
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes. 
 
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes. 
 
Find the total number of pieces of candy given to the children who always tell the truth.

Answer: 7
user: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. Find the value of $\cos(\angle CMD)$. The original answer is in the form of \frac{k}{m}, where k and m are integers in simplest form. Please provide the value of k + m.

Answer: 4
user: Kayla rolls four fair $6$-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 142
user: Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ Find the length of $AD.$

Answer: 10
user: Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. Find the integer closest to the area of $\mathcal{R}$.

Answer: 13
user: How many positive integers can fill the blank in the sentence below?

“One positive integer is _____ more than twice another, and the sum of the two numbers is $28$.” Provide your answer as an integer.

Answer: 9
user: What is the integer value of the expression:
$$(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)$$
where $\log$ denotes the base-ten logarithm?

Answer: 2
user: The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. Find $\tan(\angle ABC)$. The original answer is in the form \frac{k}{m}, please give the value of k + m.

Answer: 11
user: In how many ways can the letters in $BEEKEEPER$ be rearranged so that two or more $E$s do not appear together? Provide the number of such arrangements.

Answer: 24
user: Quadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. Find the value of $a+b+c$.

Answer: 1565
user: Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is the value of $S_{20}$?

Answer: 400
user: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?

Answer: 30
user: The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. Find the area of the region inside both the square and the rectangle. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 133
user: A $\emph{triangular number}$ is a positive integer that can be expressed in the form $t_n = 1+2+3+\cdots+n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?

Answer: 18
user: Suppose \(x\) and \(y\) are positive real numbers such that 
\(x^y=2^{64}\) and \((\log_2{x})^{\log_2{y}}=2^{7}.\)
The original answer is in the form \(k+\sqrt{m}\). Please find the value of \(k + m\).

Answer: 5
user: How many three-digit positive integers $\underline{a} \ \underline{b} \ \underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy 
$$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$
(The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ is the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$) 
Find the number of such integers.

Answer: 13
user: Suppose \(a\) is a real number such that the equation 
\[ a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)} \]
has more than one solution in the interval \((0, \pi)\). The set of all such \(a\) can be written in the form \((p,q) \cup (q,r),\) where \(p, q,\) and \(r\) are real numbers with \(p < q < r\). What is the value of \(p+q+r\)?

Answer: -4
user: Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself.

Answer: 359
user: The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle?

Answer: 40
user: Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that 

$$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}$$ 

What is the remainder when $h$ is divided by $17$?

Answer: 5
user: 在$\triangle{ABC}$中，中线$\overline{AD}$和$\overline{BE}$在$G$点相交，且$\triangle{AGE}$是等边三角形。那么$\cos(C)$可以写成$\frac{m\sqrt{p}}{n}$的形式，其中$m$和$n$是互质的正整数，$p$是一个不被任何素数的平方整除的正整数。请计算$m+n+p$的值。

Answer: 44
user: Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. 
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points? 
(Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)

Answer: 4
user: A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$, $60$, and $91$. Find the fourth term of this sequence.

Answer: 206
user: Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$? If the answer is $x^\circ$, what is the value of $x$?

Answer: 113
user: Isosceles trapezoid \(ABCD\) has parallel sides \(\overline{AD}\) and \(\overline{BC},\) with \(BC < AD\) and \(AB = CD.\) There is a point \(P\) in the plane such that \(PA=1, PB=2, PC=3,\) and \(PD=4.\) Find the value of \(\frac{BC}{AD}.\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is in simplest form. Please give the value of \(k + m\).

Answer: 4
user: Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder is $2x+1$. There is a unique polynomial of least degree with these two properties. Find the sum of the squares of the coefficients of that polynomial.

Answer: 23
user: The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$? 
$\begin{array}{|c|c|c|}\hline-2&9&5\\\hline&&-1\\\\\hline x&&8\\\hline\end{array}$

Answer: 8
user: Let 
$P(x) = x^{2022} + x^{1011} + 1.$
Find a polynomial that is a factor of $P(x)$. The original answer is in the form ax^6 + bx^3 + c. Please provide the value of a + b + c.

Answer: 3
user: Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. If the sum of the areas of all circles in $S$ is $A\pi$, what is the value of $A$?

Answer: 136
user: Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

Answer: 9
user: Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive direction, using up \(t_n\) minutes. If the total elapsed time has exceeded \(1\) minute during the \(n\)th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most \(3\) steps in all. Find the probability that Amelia’s position when she stops will be greater than \(1\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 5
user: A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus $3$ stops behind. After how many minutes will Zia board the bus?

Answer: 17
user: Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define 
$$S_n = \sum_{k=0}^{n-1} x_k 2^k$$
Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. Find the value of the sum 
$$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}.$$

Answer: 6
user: Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, find the greatest possible value of
\[f(f(800))-f(f(400))?\]

Answer: 50
user: The figure below depicts a regular $7$-gon inscribed in a unit circle.
import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]
Find the sum of the $4$th powers of the lengths of all $21$ of its edges and diagonals.

Answer: 147
user: Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale).
size(8cm); label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow()); draw((-34,0)--(60,0),EndArrow()); [/asy]
The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?

Answer: 337
user: Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define 
$S_n = \sum\limits_{k=0}^{n-1} x_k 2^k$ 
Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. Find the value of the sum 
$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}.$

Answer: 6
user: A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops, what is the probability that the cricket has returned to the leaf where it started? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the value of \(k + m\).

Answer: 34
user: Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only \(\frac{1}{3}\) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice? Provide your answer in the form \(\frac{k}{m}\) in simplest form, and then give the value of \(k + m\).

Answer: 7
user: How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? Find the number of digits.

Answer: 18
user: Jackson's paintbrush makes a narrow strip with a width of \(6.5\) millimeters. Jackson has enough paint to make a strip \(25\) meters long. How many square centimeters of paper could Jackson cover with paint? The original answer is in the form \(a \times b\). Please calculate the value of \(a + b\).

Answer: 90
user: The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? Provide your answer as an integer.

Answer: 3
user: Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. Find the probability that at some point, her running total will equal $3$. Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and then give the value of \(k + m\).

Answer: 265
user: An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube?

Answer: 126
user: When the roots of the polynomial 

\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]

are removed from the number line, what remains is the union of $11$ disjoint open intervals. Find the number of these intervals on which \( P(x) \) is positive.

Answer: 6