ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Five persons of different heights stand next to each other on numbered booths to take a picture. In how many ways can they be arranged such that the people in positions $1$ and $3$ are both taller than the person in position $2$?
Answer: 40
Problem: Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?
Answer: 122
Problem: Let $ABC$ and $DEF$ be two triangles such that $AB = DE = 20$, $BC = EF = 13$, and $\angle A = \angle D$. If $AC - DF = 10$, determine the area of $\triangle ABC$.
Answer: 126
Problem: In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$, what is the perimeter of $\triangle ABC$?
Answer: 36
Problem: Find the remainder when $20^{20} + 21^{21} - 21^{20} - 20^{21}$ is divided by 100.
Answer: 20
Problem: On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n = 0, 1, \ldots, 7^{20} - 1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
Answer: 3680
Problem: For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.
Answer: 632
Problem: If $a, b, c$ are the roots of the polynomial $x^3 + 20x^2 + 1x + 5$, compute $(a^2 + 1)(b^2 + 1)(c^2 + 1)$.
Answer: 225
Problem: Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, the expected value of the number in Rachel's hands is $E$. Compute $\left\lfloor\frac{E}{10}\right\rfloor$.
Answer: 13
Problem: Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$.
Answer: -90
Problem: We color each number in the set $S = \{1, 2, \ldots, 61\}$ with one of $25$ given colors. It is not necessary for every color to be used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?
Answer: 119
Problem: An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. The bands are colored alternately red and blue, with the smallest band colored red. The difference between the total area in red and the total area in blue is $20\, \text{cm}^2$. 

What is the area of triangle $ABC$?
Answer: 200
Problem: Katherine makes Benj play a game called \(50\) Cent. Benj starts with \(\$0.50\), and every century thereafter has a \(50\%\) chance of doubling his money and a \(50\%\) chance of having his money reset to \(\$0.50\). What is the expected value of the amount of money Benj will have, in dollars, after \(50\) centuries?
Answer: 13
Problem: The roots of the equation $x^3 - x + 1 = 0$ are $a$, $b$, and $c$. Find $a^8 + b^8 + c^8$.
Answer: 10
Problem: Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
Answer: 40
Problem: Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EFGH$ be another regular tetrahedron such that the volume of $EFGH$ is $\frac{1}{8}$-th the volume of $ABCD$. The height of $EFGH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?
Answer: 7
Problem: Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.

Nair builds triangular figures by following these rules:
- Figure $1$ is made up of $4$ small pieces.
- Figure $2$ is made up of $2$ large pieces and $8$ small pieces.
- Figure $3$ is made up of $6$ large pieces and $12$ small pieces, and so on.
- The central column must be made up exclusively of small parts.
- Outside the central column, only large pieces can be placed.

Following the pattern, how many pieces will Nair use to build Figure $20$?
Answer: 460
Problem: An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown in the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \{1,2,3,4,5\}$.

The covering satisfies the following conditions:

1) It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid.
2) Every unit triangle in the grid is covered, no two figures overlap, and every figure is fully contained in $T$.

Determine the smallest possible number of figures of type $1$ that can be used in such a covering.
Answer: 1011
Problem: Let $\triangle ABC$ be an equilateral triangle with side length $1$ and $M$ be the midpoint of side $AC$. Let $N$ be the foot of the perpendicular from $M$ to side $AB$. The angle bisector of angle $\angle BAC$ intersects $MB$ and $MN$ at $X$ and $Y$, respectively. If the area of triangle $\triangle MXY$ is $\mathcal{A}$, and $\mathcal{A}^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Answer: 769
Problem: The straight line $y = ax + 16$ intersects the graph of $y = x^3$ at 2 distinct points. What is the value of $a$?
Answer: 12
Problem: Let $a_n$ be a sequence with $a_0 = 1$ and defined recursively by
\[
a_{n+1} = \begin{cases}
a_n + 2 & \text{if } n \text{ is even},\\
2a_n & \text{if } n \text{ is odd.}
\end{cases}
\]
What are the last two digits of $a_{2015}$?
Answer: 38
Problem: Let $AB$ be a diameter of a circle and let $C$ be a point on the segment $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $\frac{[ABD]}{[CDE]}$ to the nearest integer.
Answer: 13
Problem: The real numbers $x$ and $y$ satisfy the following system of equations:
\[
\begin{cases} 
\sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\
\sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} 
\end{cases}
\]
Find the numerical value of the ratio $\frac{y}{x}$.
Answer: 6
Problem: Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x) = (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
Answer: 545
Problem: Consider all ordered pairs \((a, b)\) of positive integers such that \(\frac{a^2 + b^2 + 2}{ab}\) is an integer and \(a \leq b\). We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus \(P_1 = (1, 1)\), \(P_2 = (1, 3)\), and so on. If \(P_{2020} = (x, y)\), then compute the remainder when \(x + y\) is divided by \(2017\).
Answer: 52
Problem: Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address but forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 5
Problem: Determine the natural number $a = \frac{p+q}{r} + \frac{q+r}{p} + \frac{r+p}{q}$ where $p$, $q$, and $r$ are prime positive numbers.
Answer: 6
Problem: Let $x$ and $y$ be integers satisfying both $x^2 - 16x + 3y = 20$ and $y^2 + 4y - x = -12$. Find $x + y$.
Answer: 18
Problem: Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$?
Answer: 2042220
Problem: For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16n)}$.
Answer: 13
Problem: How many (possibly empty) sets of lattice points $\{P_1, P_2, \ldots, P_M\}$, where each point $P_i = (x_i, y_i)$ for $x_i, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$? An infinite slope, e.g., $P_i$ is vertically above $P_j$, does not count as positive.
Answer: 3432
Problem: Let \( f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65 \).
Let the roots of \( f(x) \) be \( a_k + ib_k \) for \( k = 1, 2, 3, 4 \).
Given that the \( a_k \), \( b_k \) are all integers, find \( |b_1| + |b_2| + |b_3| + |b_4| \).
Answer: 10
Problem: For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.
Answer: 49995000
Problem: How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]?
Answer: 5
Problem: There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.
Answer: 5
Problem: Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible area of $ABCD$.
Answer: 32
Problem: Find all natural numbers $n$ for which $2^8 + 2^{11} + 2^n$ is a perfect square.
Answer: 12
Problem: How many non-empty subsets of $\{1,2,\dots,11\}$ are there such that the product of its elements is the cube of an integer?
Answer: 19
Problem: A company of \( n \) soldiers is such that:
1. \( n \) is a palindrome number (reads the same forwards and backwards).
2. When the soldiers arrange in rows of 3, 4, or 5 soldiers, the last row contains 2, 3, and 5 soldiers, respectively.

Find the smallest \( n \) satisfying these conditions and prove that there are infinitely many such numbers \( n \).
Answer: 515
Problem: How many factors of $$(20^{12})^2$$ less than $$20^{12}$$ are not factors of $$20^{12}$$?
Answer: 288
Problem: Let $S=\{1,2,3,\ldots,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
Answer: 31
Problem: For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1$, $d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.
Answer: 481
Problem: The sequence $a_n$ satisfies the equation $a_{m+n} + a_{m-n} = \frac{1}{2}(a_{2m} + a_{2n})$ for all $m \geq n \geq 0$. Given that $a_1 = 1$, find $a_{1995}$.
Answer: 3980025
Problem: Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a + b$.
Answer: 132
Problem: Let $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have?
Answer: 3225600
Problem: Starting with a list of three numbers, the "Make-My-Day" procedure creates a new list by replacing each number with the sum of the other two. For example, from \( \{1, 3, 8\} \) "Make-My-Day" gives \( \{11, 9, 4\} \) and a new "Make-My-Day" leads to \( \{13, 15, 20\} \). If we begin with \( \{20, 1, 8\} \), what is the maximum difference between two numbers on the list after 2018 consecutive "Make-My-Day"s?
Answer: 19
Problem: Alice starts with the number $0$. She can apply $100$ operations on her number. In each operation, she can either add $1$ to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?
Answer: 94
Problem: The integers $1, 2, \dots, 20$ are written on the blackboard. Consider the following operation as one step: choose two integers $a$ and $b$ such that $a - b \ge 2$ and replace them with $a - 1$ and $b + 1$. Determine the maximum number of steps that can be done.
Answer: 330
Problem: A \textit{divisibility chain} is a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that $a_k$ divides $a_{k+1}$ for all $1 \le k < n$. Compute the number of divisibility chains of the form $(a, b, a^2, c, a^3, 360^9)$.
Answer: 1617000
Problem: The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square, whose sides are diagonals drawn to these division points, has an area of $13$. Find the area of the large square.
Answer: 104
Problem: Let $r_1$, $r_2$, $r_3$ be the (possibly complex) roots of the polynomial $x^3 + ax^2 + bx + \frac{4}{3}$. How many pairs of integers $a$, $b$ exist such that $r_1^3 + r_2^3 + r_3^3 = 0$?
Answer: 3
Problem: Find the least real number $m$ such that with all $5$ equilateral triangles with a total area of $m$, we can cover an equilateral triangle with side length $1$.
Answer: 2
Problem: The sum of the decimal digits of a natural number $n$ equals 100, and the sum of the digits of $44n$ equals 800. Determine the sum of the digits of $3n$.
Answer: 300
Problem: The altitudes in triangle $ABC$ have lengths 10, 12, and 15. The area of $\triangle ABC$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers and $n$ is a positive integer not divisible by the square of any prime. Find $m + n + p$.
Answer: 254
Problem: A cuboid has an integer volume. Three of the faces have different areas, namely $7$, $27$, and $L$. What is the smallest possible integer value for $L$?
Answer: 21
Problem: Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn + 13m + 13n - m^2 - n^2$. From the value of this expression alone, he miraculously determines both $m$ and $n$. What is the value of the above expression?
Answer: 169
Problem: We define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is
\[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\]
Answer: 0
Problem: A point $P$ is given in the square $ABCD$ such that $PA=3$, $PB=7$, and $PD=5$. Find the area of the square.
Answer: 58
Problem: Alice has three daughters, each of whom has two daughters. Each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)
Answer: 2926
Problem: Triangle $ABC$ has side lengths $AB=65$, $BC=33$, and $AC=56$. Find the radius of the circle that is tangent to sides $AC$ and $BC$, and also tangent to the circumcircle of triangle $ABC$.
Answer: 24
Problem: Evin's calculator is broken and can only perform three operations:

1. Given a number $x$, output $2x$.
2. Given a number $x$, output $4x + 1$.
3. Given a number $x$, output $8x + 3$.

Initially starting with the number $0$, determine how many distinct numbers at most $128$ can be generated using these operations.
Answer: 82
Problem: Determine the number of positive integers $n$ satisfying the following conditions:

1. $n < 10^6$
2. $n$ is divisible by 7
3. $n$ does not contain any of the digits 2, 3, 4, 5, 6, 7, 8.
Answer: 104
Problem: The triangle $ABC$ satisfies $AB=10$ and has angles $\angle A=75^{\circ}$, $\angle B=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\frac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.
Answer: 29
Problem: Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and \[ g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \] holds for all nonzero real numbers $x$. Find $g(2)$.
Answer: 2047
Problem: Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$.
Answer: 1178
Problem: Consider a sequence $x_1, x_2, \cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \dots, 10$, let \[ x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}. \] Suppose $x_n > 0$ for $n = 1, 2, \dots, 11$ and $x_{12} = 0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a} + \sqrt{b}}{c}$ for positive integers $a, b, c$ with $a > b$ and no square dividing $a$ or $b$. Find $100a + 10b + c$.
Answer: 622
Problem: Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.
Answer: 504
Problem: Two perpendicular planes intersect a sphere in two circles. These circles intersect at two points, $A$ and $B$, such that $AB = 42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere.
Answer: 6831
Problem: A pharmaceutical company produces a disease test with the following characteristics:
- **Accuracy on infected individuals**: $95\%$
- **Accuracy on non-infected individuals**: $90\%$

The test is applied to a population of mathletes, where $2\%$ of them actually have an infection. If a test result indicates that a mathlete has an infection, calculate the probability that the mathlete is indeed infected. This probability is expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the sum $a + b$.
Answer: 136
Problem: In circle $\Omega$, let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$, with $C$ closer to $B$ and $D$ closer to $A$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$, have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$. Then $EF=\frac{m}{n}$, for relatively prime integers $m$ and $n$. Find $m+n$.
Answer: 961
Problem: Consider the grid of numbers shown below:
\[
\begin{array}{ccccc}
20 & 01 & 96 & 56 & 16 \\
37 & 48 & 38 & 64 & 60 \\
96 & 97 & 42 & 20 & 98 \\
35 & 64 & 96 & 40 & 71 \\
50 & 58 & 90 & 16 & 89 \\
\end{array}
\]
Among all paths that start on the top row, move only left, right, and down, and end on the bottom row, what is the minimum sum of their entries?
Answer: 196
Problem: Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.
Answer: 253
Problem: Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$.
Answer: 24
Problem: Points $A$, $B$, $C$, and $D$ lie on a circle $\Gamma$, in that order, with $AB=5$ and $AD=3$. The angle bisector of $\angle ABC$ intersects $\Gamma$ at point $E$ on the opposite side of $\overleftrightarrow{CD}$ as $A$ and $B$. Assume that $\overline{BE}$ is a diameter of $\Gamma$ and $AC=AE$. Compute $DE$.
Answer: 4
Problem: For a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$.
Answer: 2
Problem: Matt has somewhere between $1000$ and $2000$ pieces of paper that he is trying to divide into piles of the same size. He attempts to divide them into $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles, but each time he ends up with one sheet left over. How many piles does he need?
Answer: 41
Problem: How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}$?
Answer: 64
Problem: The figure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles. 

Note: The figure is a nonagon (9-sided polygon).
Answer: 252
Problem: In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$.
Answer: 252
Problem: Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of which integer?
Answer: 4
Problem: There are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them.
Answer: 9
Problem: There are $63$ houses at distances of $1, 2, 3, \ldots, 63$ km from the North Pole, respectively. Santa Claus wants to distribute vaccines to each house. To do so, he will have his $63$ assistants, elves named $E_1, E_2, \ldots, E_{63}$, deliver the vaccine to each house; each elf will deliver the vaccine to exactly one house and never return. Suppose that elf $E_n$ takes $n$ minutes to travel $1$ km for each $n = 1, 2, \ldots, 63$, and that all elves leave the North Pole simultaneously. What is the minimum amount of time to complete the delivery?
Answer: 1024
Problem: Let the rest energy of a particle be $E$. Let the work done to increase the speed of this particle from rest to $v$ be $W$. If $W = \frac{13}{40} E$, then $v = kc$, where $k$ is a constant. Find $10000k$ and round to the nearest whole number.
Answer: 6561
Problem: Let $f: \mathbb{N} \to \mathbb{R}$ be the function \[ f(n) = \sum_{k=1}^{\infty} \frac{1}{\operatorname{lcm}(k,n)^2}. \] It is well-known that $f(1) = \frac{\pi^2}{6}$. What is the smallest positive integer $m$ such that $m \cdot f(10)$ is the square of a rational multiple of $\pi$?
Answer: 42
Problem: Determine the least possible value of $f(1998)$, where $f: \mathbb{N} \to \mathbb{N}$ is a function such that for all $m, n \in \mathbb{N}$,  

$$f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}.$$
Answer: 120
Problem: A positive integer $n$ is called bad if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100$.
Answer: 30