ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Henry starts with a list of the first 1000 positive integers and performs a series of steps on the list. At each step, he erases any non-positive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
Answer: 11
Problem: Consider two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$?
Answer: 24
Problem: The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$, such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$, and point $N$ is on the circle centered at $B$, such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$.
Answer: 140
Problem: Find a sequence using all the digits from $1$ to $9$ without repetition, such that any two consecutive digits form a number divisible by $7$ or $13$.
Answer: 784913526
Problem: Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \neq A$. If $AB = 10$, $AD = 24$, $XA = 17$, and $XB = 21$, then the area of $\triangle DEY$ can be written in simplest form as $\frac{m}{n}$. Find $m + n$.
Answer: 421
Problem: Let $f(x) = x^2 + 4x + 2$. Determine the difference $r$ between the largest and smallest real solutions of the equation $f(f(f(f(x)))) = 0$. Express $r$ in the form $a^{\frac{p}{q}}$, where $a$ is square-free, and $p, q$ are relatively prime positive integers. Compute $a + p + q$.
Answer: 35
Problem: We define a number as "funny" if it is divisible by the sum of its digits plus one. For example, since $1 + 2 + 1 \mid 12$, the number $12$ is considered funny. What is the maximum number of consecutive funny numbers?
Answer: 2
Problem: Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell > m > n$ and $\{\frac{3^\ell}{10^4}\} = \{\frac{3^m}{10^4}\} = \{\frac{3^n}{10^4}\}$, where $\{x\} = x - \lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.
Answer: 3003
Problem: Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$?
Answer: 3888
Problem: Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron. This is done by cutting along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?
Answer: 974
Problem: Find the least natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \ldots$, such that:
1. $d_3 + d_4 = d_5 + 6$
2. $d_4 + d_5 = d_6 + 7$
Answer: 494
Problem: Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?
Answer: 101
Problem: Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1$, $x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.
Answer: 75
Problem: A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
Answer: 11
Problem: Gabriel plays a game of drawing triangles using the vertices of a regular polygon with $2019$ sides, following these rules:

1. The vertices used by each triangle must not have been previously used.
2. The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.

If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.
Answer: 673
Problem: Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\]
Answer: 875
Problem: Let $f(x, y) = \frac{x + y}{2}$, $g(x, y) = \sqrt{xy}$, $h(x, y) = \frac{2xy}{x + y}$, and let
$$ S = \{ (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \neq b \text{ and } f(a, b), g(a, b), h(a, b) \in \mathbb{N} \} $$
where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$.
Answer: 25
Problem: Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that \( \frac{M}{N} \) is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.
Answer: 29
Problem: $N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically, or diagonally. If there are $1020$ handshakes, what is $N$?
Answer: 280
Problem: A permutation of a finite set is a one-to-one function from the set onto itself. A cycle in a permutation $P$ is a nonempty sequence of distinct items $x_1, \ldots, x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1) = x_2$ and $P(x_2) = x_1$. Every permutation of a finite set splits the set into a finite number of disjoint cycles. If this number equals 2, then the permutation is called bi-cyclic. Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".
Answer: 1764
Problem: If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$, where $a, b, c, d$ are integers, what is the value of $|a + b + c + d|$?
Answer: 93
Problem: One rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005^{th}$ digit that you write?
Answer: 5
Problem: Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move, one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.
Answer: 16
Problem: Let $x$, $y$, and $z$ be positive integers satisfying the following system of equations:

\[
\begin{align*}
&x^2 + \frac{2023}{x} = 2y^2, \\
&y + \frac{2028}{y^2} = z^2, \\
&2z + \frac{2025}{z^2} = xy.
\end{align*}
\]

Find $x + y + z$.
Answer: 25
Problem: Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares.
Answer: 7350
Problem: Find the largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.
Answer: 996
Problem: Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$.
Answer: 36
Problem: A $9 \times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call a "special" rectangle any rectangle of $2 \times 3$, $3 \times 2$, $4 \times 5$, or $5 \times 4$ on the board. 

The permitted operations are:
- Simultaneously add $1$ to all the numbers located in a special rectangle.
- Simultaneously subtract $1$ from all numbers located in a special rectangle.

Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares have $0$ (zero). What number is left in the remaining box?
Answer: 1
Problem: What is the remainder when $128^{2023}$ is divided by $126$?
Answer: 2
Problem: There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that:

$$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x} = \frac{12}{11}$$

and:

$$\sin x + \cos x = \frac{\sqrt{m}}{n}$$

where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.
Answer: 8
Problem: Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + \cdots + \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \leq 2020$.
Answer: 1926
Problem: A positive integer $N$ has 20 digits when written in base 9 and 13 digits when written in base 27. How many digits does $N$ have when written in base 3?
Answer: 39
Problem: Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.
Answer: 505
Problem: Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative" sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$.
Answer: 605
Problem: A sequence of integers $a_1$, $a_2$, $a_3$, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$.  

What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?
Answer: 986
Problem: Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$.
Answer: 423
Problem: Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex on the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
Answer: 81
Problem: Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?
Answer: 11
Problem: The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:

"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers."

How many possibilities are there for the set of truth-tellers on the U.S. Senate?
Answer: 101
Problem: Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
Answer: 8
Problem: This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday?
Answer: 2032
Problem: How many 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) are there such that each of \(a_1,a_2,a_3,a_4,a_5,a_6\) is from the set \(\{1,2,3,4\}\) and the six expressions  
\[
a_j^2 - a_j a_{j+1} + a_{j+1}^2
\]  
for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another?
Answer: 40
Problem: A box contains red, green, blue, and white balls, totaling 111 balls. If you take out 100 balls without looking, there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to ensure that among them there will always be balls of at least 3 different colors?
Answer: 88
Problem: For real numbers $a$, $b$, and $c$, the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.
Answer: 67
Problem: Find the sum of all positive integers $B$ such that $(111)_B = (aabbcc)_6$, where $a, b, c$ represent distinct base $6$ digits, with $a \neq 0$.
Answer: 237
Problem: How many integers $n$ satisfy the conditions $1 \leq n \leq 2020$ and $n^n$ is a perfect square?
Answer: 1032
Problem: Consider a right cylinder with height $5\sqrt{3}$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the difference of the lengths of the major and minor axes of this ellipse.
Answer: 75
Problem: Points $A$, $B$, $C$, $D$, and $E$ are located on a plane. It is known that:
- $CA = 12$
- $AB = 8$
- $BC = 4$
- $CD = 5$
- $DB = 3$
- $BE = 6$
- $ED = 3$

Find the length of $AE$.
Answer: 10
Problem: Form a square with sides of length $5$. Triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer.
Answer: 4
Problem: Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?
Answer: 45
Problem: Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, and the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?
Answer: 30
Problem: Cai and Lai are eating cookies. Their cookies are in the shape of $2$ regular hexagons glued to each other, and the cookies have an area of $18$ units. They each make a cut along the $2$ long diagonals of a cookie; this now makes four pieces for them to eat and enjoy. What is the minimum area among the four pieces?
Answer: 3
Problem: Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \).

**Note:**
- The function \( d(i) \) outputs the number of divisors of \( i \).
- \( \sigma (i) \) outputs the sum of the factors of \( i \).
- \( \phi (i) \) outputs the number of positive integers less than or equal to \( i \) that are relatively prime to \( i \).
Answer: 350
Problem: Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$.
Answer: 62
Problem: The math team at Jupiter Falls Middle School meets twice a month during the summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations:

\[\begin{align*}
2x+3y+3z &= 8,\\
3x+2y+3z &= 808,\\
3x+3y+2z &= 80808.
\end{align*}\]

Their goal is not to find a solution \((x,y,z)\) to the system, but instead to compute the sum of the variables. Find the value of \(x+y+z\).
Answer: 10203
Problem: If $a$ and $b$ are natural numbers such that $a + 13b$ is divisible by $11$ and $a + 11b$ is divisible by $13$, then find the least possible value of $a + b$.
Answer: 28
Problem: Michael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order. 

Given a 7-digit number format, consider the number $987-1234$ where the first four digits $(9-8-7-1)$ descend and the last four digits $(1-2-3-4)$ ascend.

Calculate the total number of 7-digit telephone numbers possible under these conditions.
Answer: 2400
Problem: There are $256$ players in a tennis tournament ranked from $1$ to $256$, where $1$ is the highest rank and $256$ is the lowest rank. In a match, the player with the higher rank wins with a probability of $\frac{3}{5}$. 

In each round, players are paired such that the highest-ranked player faces the second-highest, the third faces the fourth, and so on. Winners proceed to the next round, while losers are eliminated. After eight rounds, one player remains and is declared the winner.

Determine the expected value of the rank of the winner.
Answer: 103
Problem: In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$ given the following conditions:
- $AB \cdot \tan(\angle BCD) = 20$
- $CD = 13$
Answer: 260
Problem: Let $n$ be an integer between $0$ and $5^5$. Arrange $n$ black stones and $5^5-n$ white stones in a row, and repeat the following operation $5$ times: group the stones in sets of $5$ from left to right, and replace the $5$ stones in each group with one stone of the color that appears more frequently among them. Find the minimum possible value of $n$ such that the last remaining stone is always black, regardless of the initial arrangement.
Answer: 2883
Problem: Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and 
\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\]
Determine $a_{14^{14}}$.
Answer: 1
Problem: A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?
Answer: 150
Problem: A BWM tree is defined recursively:

- An empty tree is a BWM tree of height 0 and size 0.
- A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
- The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.

What is the minimum size of a height-10 BWM tree?
Answer: 154
Problem: The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length $36\pi$. Find $n$ such that the area of the shaded region is $n\pi$.
Answer: 108
Problem: Find the number of $10$-tuples $(a_1,a_2,\dots,a_{10})$ of integers such that $|a_1| \leq 1$ and
\[a_1^2 + a_2^2 + a_3^2 + \cdots + a_{10}^2 - a_1a_2 - a_2a_3 - a_3a_4 - \cdots - a_9a_{10} - a_{10}a_1 = 2.\]
Answer: 3780
Problem: The polynomial $R(x)$ is the remainder when $x^{2007}$ is divided by $x^2 - 5x + 6$. Express $R(0)$ in the form $ab(a^c - b^c)$. Find the value of $a + c - b$.
Answer: 2005
Problem: Katya Ugolnikova drew several angles, each greater than $10^\circ$, less than $90^\circ$, and expressed as an integer number of degrees. What is the minimum number of drawn angles required to ensure that at least three of them are equal?
Answer: 159
Problem: Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), what is \( m+n \)?
Answer: 1151
Problem: Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|$ (where $x>0$) and the line $y=2$.
Answer: 2
Problem: Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$?
Answer: 455
Problem: Right triangular prism $ABCDEF$ has triangular faces $\triangle ABC$ and $\triangle DEF$, with edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. The angles are $\angle ABC = 90^\circ$ and $\angle EAB = \angle CAB = 60^\circ$. Given that $AE = 2$, the volume of $ABCDEF$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 5
Problem: Let $s_n$ be the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_1, a_2, a_3,\) and \(a_4\) are elements of the set \(\{2, 3, 5, 7\}\) and \(b_1\) and \(b_2\) are elements of the set \(\{1, 2, 3, 4\}\). Find the number of \(n\) for which \(s_n\) is odd.
Answer: 12
Problem: The $12$-sided polygon below was created by placing three $3 \times 3$ squares with their sides parallel so that vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided polygon.
Answer: 24
Problem: What are the last two digits of $2017^{2017}$?
Answer: 77
Problem: Considering all numbers of the form $n = \left\lfloor \frac{k^3}{2012} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
Answer: 1995
Problem: Let $A_0A_1 \dots A_{11}$ be a regular $12$-gon inscribed in a circle with diameter $1$. For how many subsets $S \subseteq \{1,\dots,11\}$ is the product $\prod_{s \in S} A_0A_s$ equal to a rational number? (The empty product is declared to be $1$.)
Answer: 192
Problem: The squares of an $8 \times 8$ board are colored alternately black and white. A rectangle consisting of some of the squares of the board is called important if its sides are parallel to the sides of the board and all its corner squares are colored black. The side lengths can be anything from $1$ to $8$ squares. On each of the $64$ squares of the board, we write the number of important rectangles in which it is contained. The sum of the numbers on the black squares is $B$, and the sum of the numbers on the white squares is $W$. Determine the difference $B - W$.
Answer: 100
Problem: A sequence \((a_n)\) of real numbers is defined by \(a_0=1\), \(a_1=2015\) and for all \(n \geq 1\), we have:
\[
a_{n+1} = \frac{n-1}{n+1}a_n - \frac{n-2}{n^2+n}a_{n-1}.
\]
Calculate the value of 
\[
\frac{a_1}{a_2} - \frac{a_2}{a_3} + \frac{a_3}{a_4} - \frac{a_4}{a_5} + \ldots + \frac{a_{2013}}{a_{2014}} - \frac{a_{2014}}{a_{2015}}.
\]
Answer: 3021
Problem: Let $S(n)$ be the sum of the digits of the positive integer $n$. Determine $S(S(S(2003^{2003})))$.
Answer: 8
Problem: A subset $B$ of the set of the first $100$ positive integers has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$?
Answer: 62