sungyub/dapo-math-17k-verl huggingface dataset:
user: A trifecta is an ordered triple of positive integers \((a, b, c)\) with \(a < b < c\) such that \(a\) divides \(b\), \(b\) divides \(c\), and \(c\) divides \(ab\). What is the largest possible sum \(a + b + c\) over all trifectas of three-digit integers?

Answer: 1736
user: Let $S_0 = 0$ and let $S_k$ equal $a_1 + 2a_2 + \ldots + ka_k$ for $k \geq 1$. Define $a_i$ to be $1$ if $S_{i-1} < i$ and $-1$ if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_k = 0$?

Answer: 1092
user: Compute the number of ordered pairs \((a,b)\) of positive integers satisfying \(a^b = 2^{100}\).

Answer: 9
user: What is the maximum possible value of $k$ for which $2013$ can be expressed as a sum of $k$ consecutive positive integers?

Answer: 61
user: For $f(x) = x^4 + |x|$, let $I_1 = \int_0^\pi f(\cos x)\, dx$ and $I_2 = \int_0^{\frac{\pi}{2}} f(\sin x)\, dx$. Find the value of $\frac{I_1}{I_2}$.

Answer: 2
user: Given an $8 \times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?

Answer: 20736
user: Let $P(x)$ be a polynomial of degree 10 with non-negative integer coefficients. The remainder when $P(x)$ is divided by $(x - 1)$ is 3. How many such polynomials are there?

Answer: 66
user: How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^n} + x + 1$ is divisible by $x^2 + x + 1$?

Answer: 49
user: Let $ABC$ be an acute triangle with side lengths $AB = 7$, $BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting the midpoint of $EX$ and $C$ is $\frac{a}{b}$, where $a, b$ are relatively prime integers. Find $a + b$.

Answer: 5
user: Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least 2 polynomials of positive degree with integer coefficients.

Answer: 1648
user: Find the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \ge 0.$

Answer: 0
user: A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.

Answer: 16
user: Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.

Answer: 4
user: Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $P$ and $Q$. The tangent line closer to $Q$ touches $\omega_1$ and $\omega_2$ at $M$ and $N$ respectively. If $PQ = 3$, $QN = 2$, and $MN = PN$, what is $QM^2$?

Answer: 5
user: Let $f: \mathbb{N} \to \mathbb{N}$ satisfy $n = \sum_{d|n} f(d)$ for all $n \in \mathbb{N}$. What is the sum of all possible values of $f(100)$?

Answer: 40
user: Let $A = \{1, 2, \ldots, 100\}$ and $f(k)$, where $k \in \mathbb{N}$, be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k) = 50$?

Answer: 6
user: Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.

Answer: 369
user: Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \( \frac{\text{gcd}(m,n)}{\text{lcm}(m,n)} \) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \( \frac{r}{s} \), compute \( r + s \).

Answer: 895
user: The sequence $\{a_n\}_{n\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$, with initial values $a_1 = 1$ and $a_2 = 8$. Another sequence, $\{b_n\}$, is defined by the recurrence relation $b_{n+2} = 3b_{n+1} - b_n$ for positive integers $n$, with initial values $b_1 = 1$ and $b_2 = 2$. Find $\gcd(a_{5000}, b_{501})$.

Answer: 89
user: Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square.

Answer: 829
user: Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible value of 
$$S = 2a + 3 + \frac{32}{(a - b)(2b + 3)^2}.$$

Answer: 8
user: A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments. This cup has height $h$ and volume $V$. A second cup, exactly the same shape as the first, is held upright inside the first cup so that its base is a distance of $\frac{h}{2}$ from the base of the first cup. The volume of liquid that will fit inside the first cup and outside the second cup can be written as $\frac{m}{n} \cdot V$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 93
user: Determine the last two digits of the number $2^5 + 2^{5^2} + 2^{5^3} + \ldots + 2^{5^{1991}}$, written in decimal notation.

Answer: 12
user: Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.

Answer: 144
user: Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.

Answer: 97
user: On $2020 \times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used.

Answer: 2020
user: A number is called *Norwegian* if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. 

(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)

Answer: 1344
user: What is the smallest possible natural number $n$ for which the equation $x^2 - nx + 2014 = 0$ has integer roots?

Answer: 91
user: How many positive integers $n \leq 100$ satisfy $\lfloor n\pi \rfloor = \lfloor (n-1)\pi \rfloor + 3$? Here $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$; for example, $\lfloor \pi \rfloor = 3$.

Answer: 86
user: Suppose that $a, b, c$ are complex numbers satisfying the system of equations: 
\[a^2 = b - c,\] 
\[b^2 = c - a,\] 
\[c^2 = a - b.\] 
Compute all possible values of $a + b + c$.

Answer: 0
user: Compute the number of positive integers that divide at least two of the integers in the set $\{1^1, 2^2, 3^3, 4^4, 5^5, 6^6, 7^7, 8^8, 9^9, 10^{10}\}$.

Answer: 22
user: Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2015 \cdot 2016$.

Answer: 47
user: Let $\triangle XOY$ be a right-angled triangle with $\angle XOY=90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Find the length $XY$ given that $XN=22$ and $YM=31$.

Answer: 34
user: A rectangular tea bag $PART$ has a logo in its interior at the point $Y$. The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$.

Answer: 78
user: The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion:
\[1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, \dots\]
Find the $2014$th number in this sequence.

Answer: 13
user: Let's call a natural number \textit{interesting} if any of its two consecutive digits form a number that is a multiple of $19$ or $21$. For example, the number $7638$ is interesting because $76$ is a multiple of $19$, $63$ is a multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers with $2022$ digits exist?

Answer: 9
user: Let $\triangle ABC$ be acute with $\angle BAC = 45^{\circ}$. Let $\overline{AD}$ be an altitude of $\triangle ABC$, let $E$ be the midpoint of $\overline{BC}$, and let $F$ be the midpoint of $\overline{AD}$. Let $O$ be the center of the circumcircle of $\triangle ABC$, let $K$ be the intersection of lines $DO$ and $EF$, and let $L$ be the foot of the perpendicular from $O$ to line $AK$. If $BL = 6$ and $CL = 8$, find $AL^2$.

Answer: 48
user: Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive integers, there exists a subset of 15 elements whose sum is divisible by 15.

Answer: 29
user: $A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?

Answer: 20
user: Determine the number of positive integers $n \leq 1000$ such that the sum of the digits of $5n$ and the sum of the digits of $n$ are the same.

Answer: 91
user: Find the product of all real $x$ for which 
\[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \]

Answer: -3
user: How many different solutions are there on the interval $[0, \pi]$ for the equation: $$6\sqrt{2} \sin x \cdot \tan x - 2\sqrt{2} \tan x + 3\sin x - 1 = 0$$?

Answer: 2
user: In the diagram, $ABCDEFG$ is a regular heptagon (a $7$-sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 727
user: Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.

Answer: 28
user: Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$.

Answer: 137
user: Five identical circles are placed in a line inside a larger circle as shown. If the shown chord has a length of $16$, find the radius of the large circle.

Answer: 10
user: A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle, and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.

Answer: 11
user: Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3 + a_2^3 + \cdots + a_{10}^3$.

Answer: 8021
user: For a positive integer $n$, consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. 

For example, if $n = 117$, it has six divisors: $1, 3, 9, 13, 39, 117$. Two of these divisors, namely $3$ and $13$, have unit digits equal to $3$. Hence, for $n = 117$, $p = 33.33\ldots$.

Find the maximum possible value of $p$ for any positive integer $n$.

Answer: 50
user: Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $\triangle XYZ$.

Answer: 7500
user: Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.

Answer: 2093124
user: Tim has a working analog 12-hour clock with two hands that run continuously. He also has a clock that runs at half the correct rate. At noon one day, both clocks show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?

Answer: 33
user: Find the minimum of $x + y + z$ where $x$, $y$, and $z$ are real numbers satisfying the following conditions:
- $x \geq 4$
- $y \geq 5$
- $z \geq 6$
- $x^2 + y^2 + z^2 \geq 90$

Answer: 16
user: Andy has 2010 square tiles, each with a side length of one unit. He plans to arrange the tiles in an $m \times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all the different possible rectangles he can make. Two rectangles are considered the same if one can be rotated to become the other. For example, a $1 \times 2010$ rectangle is considered the same as a $2010 \times 1$ rectangle.

Answer: 9792
user: How many divisors of $30^{2003}$ are there which do not divide $20^{2000}$?

Answer: 8044086060
user: We color all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue. Each vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same color. Suppose there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?

Answer: 859
user: In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, the segment $BC$ is divided into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. Determine the smallest possible value of $n$.

Answer: 27
user: Let $x_1$, $x_2$, \ldots, $x_{10}$ be 10 numbers. Suppose that $x_i + 2x_{i+1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512x_{10}$?

Answer: 171
user: What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6?

Answer: 576
user: How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.

Answer: 114
user: Compute $ax^{2018} + by^{2018}$, given that there exist real numbers $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\[
\begin{align*}
ax^{2014} + by^{2014} &= 6, \\
ax^{2015} + by^{2015} &= 7, \\
ax^{2016} + by^{2016} &= 3, \\
ax^{2017} + by^{2017} &= 50.
\end{align*}
\]

Answer: -417
user: Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.

Answer: 4
user: In how many rearrangements of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ do the numbers form a \textit{hill}? A \textit{hill} is defined as a sequence where the numbers increase to a peak and then decrease, such as in $129876543$ or $258976431$.

Answer: 254
user: Given that the base-17 integer $\overline{8323a02421_{17}}$ (where $a$ is a base-17 digit) is divisible by $16_{10}$, find $a$. Express your answer in base $10$.

Answer: 7
user: In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $\triangle ABC$. The circumcircles of $\triangle BFI$ and $\triangle CEI$ intersect at $X \neq I$. Find the length of $DX$.

Answer: 3
user: Trapezoid $ABCD$ is an isosceles trapezoid with $AD = BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid?

Answer: 242
user: Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.

Answer: 13
user: Triangle $ABC$ has $\overline{AB} = 5$, $\overline{BC} = 4$, $\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$, and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?

Answer: 255
user: Let $ABCD$ be a unit square. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$ respectively. Let $P$ and $Q$ be the midpoints of line segments $AM$ and $BN$ respectively. Find the reciprocal of the area of the triangle enclosed by the three line segments $PQ$, $MN$, and $DB$.

Answer: 128
user: For positive integers $m$ and $n$, the decimal representation for the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. Find the least possible value for $n$.

Answer: 45
user: Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$

Answer: 25
user: Compute the sum:
\[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \]
where every three consecutive \( + \) are followed by two \( - \).

Answer: 401799
user: If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.

Answer: 53
user: Let $u=\{a_1, \ldots, a_5\}$, where all $a_i \in \{0,1\}$. Let $D(u)$ be the set of all subsequences of $u$ with length 4. Let $A$ be a set of sequences $u_1, \ldots, u_{|A|}$ with the property that $|D(u_i) \cap D(u_j)| = 0$ for all $i \neq j$. What is the maximum value of $|A|$?

Answer: 5
user: Let $ABC$ be a triangle with incenter $I$ such that $AB = 20$ and $AC = 19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA = IP = IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.

Answer: 3902
user: A computer outputs the values of the expression $(n+1) \cdot 2^n$ for $n = 1, n = 2, n = 3$, etc. What is the largest number of consecutive values that are perfect squares?

Answer: 2
user: For any positive integer $n$, define 
\[ E(n) = n(n+1)(2n+1)(3n+1)\cdots (10n+1). \]
Find the greatest common divisor of $E(1), E(2), E(3), \dots, E(2009)$.

Answer: 2310
user: The number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?

Answer: 3004
user: Find the unique 3-digit number $N=\underline{A}\ \underline{B}\ \underline{C}$, whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}\ \underline{B}\ \underline{C} \times \underline{A}\ \underline{B} \times \underline{A}$ is divisible by $1000$.

Answer: 875
user: Giraldo wrote five distinct natural numbers on the vertices of a pentagon. Next, he wrote on each side of the pentagon the least common multiple (LCM) of the numbers written on the two vertices of that side. He noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?

Answer: 30
user: Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 109
user: Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 193
user: There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?

Answer: 126
user: Right triangle $\triangle ABC$ with a right angle at $A$ has $AB = 20$ and $AC = 15$. Point $D$ is on $AB$ with $BD = 2$. Points $E$ and $F$ are placed on ray $CA$ and ray $CB$, respectively, such that $CD$ is a median of $\triangle CEF$. Find the area of $\triangle CEF$.

Answer: 54
user: Find the smallest positive integer that is relatively prime to each of $2$, $20$, $204$, and $2048$.

Answer: 1
user: We say that a $2023$-tuple of nonnegative integers $(a_1,\ldots,a_{2023})$ is \textit{sweet} if the following conditions hold:

- $a_1+a_2+\ldots+a_{2023}=2023$
- $\frac{a_1}{2}+\frac{a_2}{2^2}+\ldots+\frac{a_{2023}}{2^{2023}}\le 1$

Determine the greatest positive integer $L$ so that $a_1+2a_2+\ldots+2023a_{2023}\ge L$ holds for every sweet $2023$-tuple $(a_1,\ldots,a_{2023})$.

Answer: 22228
user: Find the 3-digit positive integer that has the most divisors.

Answer: 840
user: Compute the number of ordered quintuples of nonnegative integers \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_1, a_2, a_3, a_4, a_5 \leq 7\) and \(5\) divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).

Answer: 6528
user: Consider the numbers $\{24, 27, 55, 64, x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$.

Answer: 60
user: A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?

Answer: 48
user: Define a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.

Answer: 160402
user: Using each of the eight digits $1, 3, 4, 5, 6, 7, 8,$ and $9$ exactly once, form a three-digit number $A$, two two-digit numbers $B$ and $C$ (where $B < C$), and a one-digit number $D$. The numbers are such that $A + D = B + C = 143$. In how many ways can this be done?

Answer: 24
user: Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers: $x$, $y$, and $10x+y$, where $x$ and $y$ are digits. What is the sum of the digits of $n$?

Answer: 12
user: A regular $2015$-simplex $\mathcal{P}$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal{P}$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal{P}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$.

Answer: 321