ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We define a pair of players $A$ and $B$ as a \textit{dominant pair} if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
Answer: 2015
Problem: Find the largest prime factor of \( \frac{1}{\sum_{n=1}^\infty \frac{2012}{n(n+1)(n+2)\cdots(n+2012)}} \).
Answer: 2011
Problem: If $xy:yz:zx = 6:8:12$, and the ratio $\frac{x^3+y^3+z^3}{xyz}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Answer: 41
Problem: Calvin makes a number. He starts with $1$, and on each move, he multiplies his current number by $3$, then adds $5$. After $10$ moves, find the sum of the digits (in base $10$) when Calvin's resulting number is expressed in base $9$.
Answer: 21
Problem: Consider all cubic polynomials $f(x)$ such that:
1. $f(2018) = 2018$,
2. The graph of $f$ intersects the $y$-axis at height 2018,
3. The coefficients of $f$ sum to 2018,
4. $f(2019) > 2018$.

We define the infimum of a set $S$ as follows: Let $L$ be the set of lower bounds of $S$. That is, $\ell \in L$ if and only if for all $s \in S$, $\ell \leq s$. Then the infimum of $S$ is $\max(L)$.

Of all such $f(x)$, what is the infimum of the leading coefficient (the coefficient of the $x^3$ term)?
Answer: 0
Problem: 2021 people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up 1000 positions to the left of their original position.
Answer: 1021000
Problem: Let $p$ be a prime and $a$ and $n$ be natural numbers such that \( \frac{p^a - 1}{p-1} = 2^n \). Find the number of natural divisors of $na$.
Answer: 4
Problem: Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$.
Answer: 287
Problem: Determine \[ \left\lfloor \prod_{n=2}^{2022} \frac{2n+2}{2n+1} \right\rfloor, \] given that the answer is relatively prime to 2022.
Answer: 29
Problem: Let $n$ be the largest number such that 
\[ \frac{2014^{100!} - 2011^{100!}}{3^n} \]
is still an integer. Compute the remainder when $3^n$ is divided by $1000$.
Answer: 83
Problem: We call a number greater than $25$ a \textit{semi-prime} if it is the sum of two different prime numbers. What is the greatest number of consecutive natural numbers that can be \textit{semi-prime}?
Answer: 5
Problem: Find the smallest prime number that cannot be written in the form $|2^a - 3^b|$ with non-negative integers $a, b$.
Answer: 41
Problem: Let $x_0, x_1, x_2, \ldots$ be the sequence defined by:  
\[x_i = 2^i \text{ if } 0 \leq i \leq 2003\]  
\[x_i = \sum_{j=1}^{2004} x_{i-j} \text{ if } i \geq 2004\]  
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
Answer: 2003
Problem: A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the midpoint of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ is the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$?
Answer: 18
Problem: Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?
Answer: 18
Problem: Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?
Answer: 150
Problem: Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that 
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If 
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Answer: 201925
Problem: Triangle $ABC$ has $AB = 4$, $BC = 5$, and $CA = 6$. Points $A'$, $B'$, $C'$ are such that $B'C'$ is tangent to the circumcircle of $\triangle ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.
Answer: 5
Problem: Let $x, y, z > 0$ such that \((x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{91}{10}\). Compute \(\left[(x^3+y^3+z^3)\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)\right]\) where \([.]\) represents the integer part.
Answer: 9
Problem: An alien script has $n$ letters $b_1, b_2, \dots, b_n$. For some $k < \frac{n}{2}$, assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. A $k$-word is considered **sacred** if:

1. No letter appears twice, and
2. If a letter $b_i$ appears in the word, then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$).

For example, if $n = 7$ and $k = 3$, then $b_1b_3b_6$, $b_3b_1b_6$, $b_2b_4b_6$ are sacred 3-words. On the other hand, $b_1b_7b_4$, $b_2b_2b_6$ are not sacred.

What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.
Answer: 600
Problem: If there is only \(1\) complex solution to the equation \(8x^3 + 12x^2 + kx + 1 = 0\), what is \(k\)?
Answer: 6
Problem: Let $\triangle ABC$ be an isosceles triangle with $AB = AC$ and $BC = 14$. This triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD = 6$. The intersection of line $AD$ with the circle $\omega$ is point $E$. Given that $AE = 7$, find $AC^2$.
Answer: 105
Problem: Compute the number of nonempty subsets $S$ of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max S + \min S}{2}$ is an element of $S$.
Answer: 234
Problem: How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points?
Answer: 36
Problem: Compute the sum of $x^2 + y^2$ over all four ordered pairs $(x, y)$ of real numbers satisfying $x = y^2 - 20$ and $y = x^2 + x - 21$.
Answer: 164
Problem: A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, such that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ is $r$. If $r = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$.
Answer: 11
Problem: A set $S$ of positive integers is \textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.

Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.
Answer: 223
Problem: Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 12
Problem: Determine the value of the series $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \ldots + 10000$, where the signs change after each perfect square.
Answer: 1000000
Problem: We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relatively prime. Find the maximum value of $n$.
Answer: 25
Problem: A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. 

Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.
Answer: 7
Problem: In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$?

$$
\begin{array}{ccccccc}
A & B & C & D & E & D & B \\
& B & C & D & E & D & B \\
&& C & D & E & D & B \\
&&& D & E & D & B \\
&&&& E & D & B \\
&&&&& D & B \\
+&&&&&& B \\
\hline
A & A & A & A & A & A & A
\end{array}
$$
Answer: 84269
Problem: Find the largest positive integer $n$ such that $n$ is 167 times the number of its positive divisors.
Answer: 2004
Problem: Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.
Answer: 50308
Problem: Find the smallest perfect square whose last digit is not $0$, such that when its last two digits are removed, the resulting number is also a perfect square.
Answer: 121
Problem: Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess:

- A digit is highlighted blue if it is in the Primle but not in the correct place.
- A digit is highlighted orange if it is in the Primle and is in the correct place.
- A digit is left unhighlighted if it is not in the Primle.

If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?

$$
\begin{array}{c} 
    \boxed{1} \quad \boxed{3} \\
    \boxed{4} \quad \fcolorbox{black}{blue}{\color{white}7}
\end{array}
$$
Answer: 79
Problem: An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
Answer: 56
Problem: Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, allowing Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$ and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes:
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]
Out of all $\binom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
Answer: 125
Problem: A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?
Answer: 2892
Problem: Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} = \frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.
Answer: 14
Problem: Find $n$ such that $\frac{1}{2!9!} + \frac{1}{3!8!} + \frac{1}{4!7!} + \frac{1}{5!6!} = \frac{n}{10!}$.
Answer: 92
Problem: A total fixed amount of $N$ thousand rupees is given to $A$, $B$, and $C$ every year, each receiving an amount proportional to their age. In the first year, $A$ received half of the total amount. When the sixth payment was made, $A$ received six-sevenths of the amount she received in the first year; $B$ received 1000 Rs less than she received in the first year, and $C$ received twice the amount she received in the first year. Find $N$.
Answer: 35
Problem: Square $ABCD$ has side length $7$. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. Similarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares.
Answer: 98
Problem: Let $x$ and $y$ be two real numbers such that \( 2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7 \). Find \( \tan^2 x + 2 \tan^2 y \).
Answer: 9
Problem: How many integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?
Answer: 21
Problem: $M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC$. The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1$. On the line passing through $C$ and parallel to $MD$, there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150^\circ$. Find the $\angle MTD$.
Answer: 120
Problem: A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$
Answer: 51
Problem: Triangle $ABC$ has side lengths $AB=20$, $AC=14$, and $BC=22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let \( \frac{p}{q} = \frac{[AMPN]}{[ABC]} \) for positive integers \( p \) and \( q \) that are coprime. Note that \([ABC]\) denotes the area of triangle $ABC$. Find \( p+q \).
Answer: 331
Problem: Evaluate the sum of all positive irreducible fractions less than $1$ with the denominator $1991$.
Answer: 900
Problem: A positive integer $n$ is called \textit{un-two} if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of \[ \frac{7a+b}{n}, \frac{7b+c}{n}, \frac{7c+a}{n} \] are integers. Find the sum of all un-two positive integers.
Answer: 660
Problem: Let $N_6$ be the answer to problem 6.

Given positive integers $n$ and $a$, the $n$th tetration of $a$ is defined as:
\[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \]

For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$.

Compute the units digit of $^{2024}N_6$.
Answer: 6
Problem: The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.

```latex
\begin{asy}
  unitsize(5);
  pair A = (0, 20 * 21) / 29.0;
  pair B = (-20^2, 0) / 29.0;
  pair C = (21^2, 0) / 29.0;
  draw(A -- B -- C -- cycle);
  label("$A$", A, S);
  label("$B$", B, S);
  label("$C$", C, S);
  filldraw(arc((A + C)/2, C, A)--cycle, gray);
  filldraw(arc((B + C)/2, C, A)--cycle, white);
  filldraw(arc((A + B)/2, A, B)--cycle, gray);  
  filldraw(arc((B + C)/2, A, B)--cycle, white);
\end{asy}
```

If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?
Answer: 210
Problem: In triangle $ABC$, we have $AB = 7$, $AC = 9$, and $BC = 8$. The angle bisector of $\angle BAC$ intersects side $BC$ at point $D$, and the angle bisector of $\angle ABC$ intersects $AD$ at point $E$. Compute $AE^2$.
Answer: 21
Problem: Find the number of subsets $S$ of $\{1, 2, \dots, 63\}$ such that the sum of the elements in $S$ is 2008.
Answer: 6
Problem: An *up-right path* from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_1), \dots, (x_k, y_k)$ of points in $\mathbb{R}^2$ such that $(a, b) = (x_1, y_1)$, $(c, d) = (x_k, y_k)$, and for each $1 \le i < k$, we have that either $(x_{i+1}, y_{i+1}) = (x_i + 1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.\n\nFind the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.
Answer: 1750
Problem: Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$. Note that the first sequence starts at $99$, and the second sequence ends at $98$.
Answer: 9801
Problem: Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
Answer: 7
Problem: A Mediterranean polynomial has only real roots and is of the form:
\[ P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \]
with real coefficients $a_0, \ldots, a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
Answer: 11
Problem: Let $a, b$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}$ and $(a-b)^2 = 4(ab)^3$. Find $\log_a b$.
Answer: -1
Problem: For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that
\[
\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N
\]
find the sum of the squares of the digits of $N$.
Answer: 56
Problem: Find the largest positive integer that cannot be expressed in the form $a + bc$, where $a$, $b$, and $c$ are positive integers satisfying $a < b < c$.
Answer: 24
Problem: Find the number of $4$-digit positive integers $n$ such that the largest power of 2 that divides $n!$ is $2^{n-1}$.
Answer: 4
Problem: Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
Answer: 48
Problem: Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.
Answer: 5792
Problem: Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?

\begin{asy}
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
\end{asy}
Answer: 42
Problem: Determine the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.
Answer: 24
Problem: Find the product of all values of $d$ such that $x^{3} + 2x^{2} + 3x + 4 = 0$ and $x^{2} + dx + 3 = 0$ have a common root.
Answer: 1
Problem: Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 421
Problem: Find the minimum possible least common multiple of twenty natural numbers whose sum is $801$.
Answer: 42
Problem: Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2 + ax + b$ and $x^2 + bx + a$ has two distinct real roots, and the product of these trinomials has exactly three distinct real roots. Determine all possible values of the sum of these three roots.
Answer: 0
Problem: Let $ABCD$ be an isosceles trapezoid with base $AD$ measuring $11$ cm and the other three sides each measuring $5$ cm. Draw a line perpendicular to $BD$ passing through $C$, and another line perpendicular to $AC$ passing through $B$. Let $E$ be the intersection point of these two lines. Determine the distance between point $E$ and line $AD$.
Answer: 1
Problem: Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$.
Answer: 210
Problem: Let $b$ be a real number randomly selected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $\frac{m}{n}$ is the probability that the equation $$x^4 + 25b^2 = (4b^2 - 10b)x^2$$ has \textit{at least} two distinct real solutions. Find the value of $m+n$.
Answer: 63
Problem: Find all three-digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! \times b! \times c!$.
Answer: 432
Problem: In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$.
Answer: 40
Problem: A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be \( \frac{m}{n} \) where \( m, n \) are numbers such that \( m, n \) are positive integers and \( \gcd(m, n) = 1 \).

Take this answer and add the numerator and denominator.
Answer: 3004
Problem: For a positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$. For example, $f_{1}(123) = (1+2+3)^{2} = 36$. Define $f_{n+1}(k) = f_{1}(f_{n}(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim.
Answer: 169
Problem: Find the integer represented by $\left\lfloor \sum_{n=1}^{10^9} n^{-2/3} \right\rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
Answer: 2997
Problem: Georg is putting his $250$ stamps in a new album. On the first page, he places one stamp, and then on every subsequent page, he places either the same number of stamps or twice as many as on the preceding page. In this way, he ends up using precisely all $250$ stamps. How few pages are sufficient for him?
Answer: 8
Problem: Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\[
\begin{align*}
1+1+1+1&=4,\\
1+3&=4,\\
3+1&=4.
\end{align*}
\]
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$.
Answer: 71
Problem: The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$.
Answer: 5
Problem: The function $f$ is defined by the following table:

\[
\begin{array}{|c||c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
f(x) & 4 & 1 & 3 & 5 & 2 \\
\hline
\end{array}
\]

Given $u_0 = 4$ and the recurrence relation $u_{n+1} = f(u_n)$ for $n \geq 0$, find $u_{2002}$.

\[
\textbf{(A) } 1 \quad \textbf{(B) } 2 \quad \textbf{(C) } 3 \quad \textbf{(D) } 4 \quad \textbf{(E) } 5
\]
Answer: 2
Problem: Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
Answer: 61