ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $u_n$ be the $n^{\text{th}}$ term of the sequence:
\[
1, \quad 2, \quad 5, \quad 6, \quad 9, \quad 12, \quad 13, \quad 16, \quad 19, \quad 22, \quad 23, \ldots,
\]
where the first term is the smallest positive integer that is $1$ more than a multiple of $3$, the next two terms are the next two smallest positive integers that are each two more than a multiple of $3$, the next three terms are the next three smallest positive integers that are each three more than a multiple of $3$, the next four terms are the next four smallest positive integers that are each four more than a multiple of $3$, and so on:
\[
\underbrace{1}_{1\text{ term}}, \quad \underbrace{2, \quad 5}_{2\text{ terms}}, \quad \underbrace{6, \quad 9, \quad 12}_{3\text{ terms}}, \quad \underbrace{13, \quad 16, \quad 19, \quad 22}_{4\text{ terms}}, \quad \underbrace{23, \ldots}_{5\text{ terms}}, \ldots.
\]
Determine $u_{2008}$.
Answer: 5898
Problem: Find the number of ordered pairs \((a, b)\) of positive integers that are solutions of the following equation: 
\[a^2 + b^2 = ab(a+b).\]
Answer: 1
Problem: Let $f : \mathbb{R} \to \mathbb{R}$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.
Answer: 849
Problem: Jim has two fair 6-sided dice: one die has faces labelled from 1 to 6, and the second die has faces labelled from 3 to 8. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given that the sum of the resulting two rolls is 9, if \( \frac{m}{n} \) is the probability he rolled the same die twice where \( m, n \) are relatively prime positive integers, then what is \( m + n \)?
Answer: 7
Problem: Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p > \sqrt{n}$ and $p < 20$. What is the value of $N$?
Answer: 69
Problem: The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation:
\[ 2x^2 + 4xy + 3y^2 - 2xz - 2yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for $t$.
Answer: 125
Problem: How many functions $f : \{1, 2, \ldots, 16\} \to \{1, 2, \ldots, 16\}$ have the property that $f(f(x)) - 4x$ is divisible by $17$ for all integers $1 \le x \le 16$?
Answer: 16
Problem: Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece, he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece, he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 67
Problem: In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$.
Answer: 3
Problem: Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
Answer: 5
Problem: What is the smallest integer \( a \) greater than 14 such that a triangle with side lengths \( a - 1 \), \( a \), and \( a + 1 \) has an integer area?
Answer: 52
Problem: A rhombus with sidelength $1$ has an inscribed circle with radius $\frac{1}{3}$. If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive integers $a$ and $b$, evaluate $a+b$.
Answer: 5
Problem: What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.
Answer: 987654301
Problem: In triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\triangle ABC$ is centered at $I$ and touches $AB$, $AC$, and $BC$ at points $M$, $N$, and $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $\triangle MAN$ and $\triangle KID$ intersects lines $AB$ and $AC$ at points $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.
Answer: 720
Problem: Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segments $EF$ and $CF$, respectively, such that $A$, $P$, and $Q$ are collinear. Given that $EP = 5$, $PF = 3$, and $QF = 12$, find $CQ$.
Answer: 8
Problem: Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote?
Answer: 16
Problem: Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB = 60^{\circ}$, $\angle BOC = 90^{\circ}$, and $\angle COA = 120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2 \theta = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m + n$.
Answer: 109
Problem: Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?

Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.
Answer: 30
Problem: Let $n$ be the smallest integer such that the sum of digits of $n$ is divisible by $5$, and the sum of digits of $(n+1)$ is also divisible by $5$. What are the first two digits of $n$ in the same order?
Answer: 49
Problem: Find the largest two-digit number $N$ which is divisible by 4, such that all integral powers of $N$ end with $N$.
Answer: 76
Problem: Aidan rolls a pair of fair, six-sided dice. Let $n$ be the probability that the product of the two numbers on top is prime. Given that $n$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a + b$.
Answer: 7
Problem: Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x - |x^2 - 4x + 3| = k.$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 11
Problem: Let $x, y \in \mathbb{R}$ such that $x > 2, y > 3$. Find the minimum value of \( \frac{(x+y)^2}{\sqrt{x^2-4} + \sqrt{y^2-9}} \).
Answer: 10
Problem: $p, q, r$ are real numbers satisfying the following equations:  
\[ \frac{(p+q)(q+r)(r+p)}{pqr} = 24 \]  
\[ \frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10. \]  
Given that \( \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \) can be expressed in the form \( \frac{m}{n} \), where \( m, n \) are relatively prime positive integers, compute \( m+n \).
Answer: 67
Problem: Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\frac{n}{3}$. Find the smallest possible value of the sum of the digits of $n$.
Answer: 126
Problem: Given two real numbers $a$ and $b$ that satisfy the equations:
\[
a^4 + a^2b^2 + b^4 = 900,
\]
\[
a^2 + ab + b^2 = 45.
\]
Find the value of $2ab$.
Answer: 25
Problem: How many integer solutions exist that satisfy the equation?

$$x + 4y - 343\sqrt{x} - 686\sqrt{y} + 4\sqrt{xy} + 2022 = 0$$
Answer: 173
Problem: Given two non-negative integers $m > n$, let's say that $m$ ends in $n$ if we can obtain $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.

Determine how many three-digit numbers end in the product of their digits.
Answer: 103
Problem: How many sets of positive integers \((a, b, c)\) satisfy \(a > b > c > 0\) and \(a + b + c = 103\)?
Answer: 833
Problem: In triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\mathcal{P}$ be the parabola with focus $A$ passing through $B$ and $C$, and denote by $T$ the intersection point of $AD$ with the directrix of $\mathcal{P}$. Determine the value of $DT^2 - DA^2$. 

(Recall that a parabola $\mathcal{P}$ is the set of points which are equidistant from a point, called the *focus* of $\mathcal{P}$, and a line, called the *directrix* of $\mathcal{P}$.)
Answer: 96
Problem: Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.
Answer: 113
Problem: What is the largest two-digit integer for which the product of its digits is $17$ more than their sum?
Answer: 74
Problem: In the unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, $\overline{DH}$ are edges of the cube), $L$ is the midpoint of $GH$. The area of $\triangle CAL$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 7
Problem: Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 843
Problem: How many distinct four-letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements.
Answer: 72
Problem: A mason has bricks with dimensions $2 \times 5 \times 8$ and other bricks with dimensions $2 \times 3 \times 7$. She also has a box with dimensions $10 \times 11 \times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.

Find all possible values of the total number of bricks that she can pack.
Answer: 24
Problem: Let $\{a_n\}$ be a sequence of integers such that $a_1=2016$ and \[\frac{a_{n-1}+a_n}{2}=n^2-n+1\] for all $n\geq 1$. Compute $a_{100}$.
Answer: 7986
Problem: Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, \dots, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.
Answer: 21
Problem: Square trinomial $f(x)$ is such that the polynomial $f(x)^5 - f(x)$ has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.
Answer: 0
Problem: The following number is the product of the divisors of $n$:
$$46,656,000,000$$
What is $n$?
Answer: 60
Problem: Define $\{p_n\}_{n=0}^\infty\subset\mathbb{N}$ and $\{q_n\}_{n=0}^\infty\subset\mathbb{N}$ to be sequences of natural numbers as follows:

- $p_0 = q_0 = 1$;
- For all $n \in \mathbb{N}$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n, q_n) = 1$ satisfying 

\[
\frac{p_{n-1}}{q_{n-1}} < \frac{p_n}{q_n} < \sqrt{2}.
\]

Find $q_3$.
Answer: 17
Problem: Let $N$ be the number of functions $f: \{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1 \leq x \leq 5$ it is true that $f(f(x)) = x$. Given that $N$ can be written in the form $5^a \cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a + b$.
Answer: 31
Problem: How many 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers with $a < b < c$ and $c = 8000$?
Answer: 39
Problem: What is the largest number of points that can be arranged on a given circle such that in all the triangles they form, none of the angles is less than $25^\circ$?
Answer: 7
Problem: Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[ N = 1111\cdots 11. \] Find the thousandth digit after the decimal point of $\sqrt{N}$.
Answer: 1
Problem: Let $BCDE$ be a trapezoid with $BE \parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be the intersection of $AM$ and $BE$. If $BX$ can be written as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, find $a + b$.
Answer: 203
Problem: Compute $\sum_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$.

Express your answer in simplest form as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
Answer: 21
Problem: Let $M$ be the set of all polynomials $P(x)$ with pairwise distinct integer roots, integer coefficients, and all absolute values of the coefficients less than $2007$. What is the highest degree among all the polynomials in the set $M$?
Answer: 9
Problem: Compute the largest integer \( N \leq 2012 \) with four distinct digits.
Answer: 1987
Problem: A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room.
Answer: 18
Problem: Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter such that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\frac{17}{128}$. There are relatively prime positive integers $m$ and $n$ such that the ratio of $DE$ to $AB$ is $\frac{m}{n}$. Find $m + n$.
Answer: 47
Problem: In Occidentalia, there are $20$ different companies, each looking to hire $15$ new employees. A group of $300$ applicants interview with each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, such that each company finds exactly $p$ suitable applicants, with $p > 15$. Furthermore, each applicant is found suitable by at least one company. What is the smallest value of $p$ for which it is always possible to assign $15$ applicants to each company, given that each company is assigned only applicants that it considers suitable, and that each of the $300$ applicants is assigned to a company?
Answer: 20
Problem: For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\} = x - \lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying:  
\[\{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20}\]  
If $S$ is the sum of all numbers in $A$, find $\lfloor S \rfloor$.
Answer: 21
Problem: The area of the largest square that can be inscribed in a regular hexagon with side length $1$ can be expressed as $a - b\sqrt{c}$, where $c$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 21
Problem: Consider two geometric sequences $16$, $a_1$, $a_2$, $\ldots$ and $56$, $b_1$, $b_2$, $\ldots$ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.
Answer: 490
Problem: How many positive integer solutions does the equation have:
\[ \left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1 \]
where \( \lfloor x \rfloor \) denotes the integer part of \( x \), for example, \( \lfloor 2 \rfloor = 2 \), \( \lfloor \pi \rfloor = 3 \), \( \lfloor \sqrt{2} \rfloor = 1 \).
Answer: 110
Problem: Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?
Answer: 1234
Problem: Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$, which is the greatest integer not exceeding $\mathcal{A}$.
Answer: 157
Problem: There are $5$ dogs, $4$ cats, and $7$ bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are identical. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are used?
Answer: 20
Problem: Let \( f(x) \) be a monic cubic polynomial with \( f(0) = -64 \), and all roots of \( f(x) \) are non-negative real numbers. What is the largest possible value of \( f(-1) \)?
Answer: -125
Problem: What number is nine more than four times the answer to this question?
Answer: -3
Problem: You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $1$ m/s. Thereafter, when exactly $1$ second has passed from when Al last changed its speed, Al’s speed instantaneously becomes $1$ m/s faster, so that Al runs at a constant speed of $k$ m/s in the $k$th second of the race. (Start counting seconds at $1$). Suppose Al beats you by exactly $1$ second. How many meters was the race?
Answer: 210
Problem: Let $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2+\dots+a_{2013}$ is divided by $1000$.
Answer: 693
Problem: How many ways are there to permute the letters $\{S, C, R, A, M, B, L, E\}$ without the permutation containing the substring $LAME$?
Answer: 40200
Problem: \( f \) is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. \( f \) satisfies the condition that \( f(f(n)) + f(n) = 2n + 3 \) for all nonnegative integers \( n \). Find \( f(2014) \).
Answer: 2015
Problem: For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[ \sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)}, \] where $N=6^{1999}$.
Answer: 8
Problem: Let $x_n = 2^{2^n} + 1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}$. Find the last digit of $m$.
Answer: 9
Problem: Juan makes a list of $2018$ numbers. The first number is $1$. Each subsequent number is obtained by adding one of the numbers $1, 2, 3, 4, 5, 6, 7, 8,$ or $9$ to the previous number. Given that none of the numbers in the list ends in $0$, what is the largest possible value for the last number on the list?
Answer: 17929
Problem: Compute the three-digit number that satisfies the following properties:
- The hundreds digit and ones digit are the same, but the tens digit is different.
- The number is divisible by $9$.
- When the number is divided by $5$, the remainder is $1$.
Answer: 171
Problem: The grid below contains five rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance of one unit apart. Find the area of the pentagon shown.

\[\text{[asy]}\]
\text{size}(150);\text{defaultpen}(\text{linewidth}(0.9));
\text{for}(\text{int } i=0; i \leq 5; ++i){
  \text{for}(\text{int } j=0; j \leq 4; ++j){
    \text{dot}((i,j));
  }
}
\text{draw}((3,0)--(0,1)--(1,4)--(4,4)--(5,2)--\text{cycle});
\[\text{[/asy]}\]
Answer: 14
Problem: Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2, \ldots, \epsilon_8)$ such that $\epsilon_1, \epsilon_2, \ldots, \epsilon_8 \in \{1, -1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 + \cdots + 8\epsilon_8$ is a multiple of $3$.
Answer: 88
Problem: In the regular hexagon $ABCDEF$, a point $X$ is marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$?
Answer: 25
Problem: Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the sum of all possible values of $n$.
Answer: 89
Problem: Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$.
Answer: 292
Problem: Let Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, "If I give you some marbles then you will have twice as many marbles as I will have." Birbal says to Akbar, "If I give you some marbles then you will have thrice as many marbles as I will have." What is the minimum possible value of $n$ for which the above statements are true?
Answer: 12
Problem: For each positive integer $n \ge 2$, define $k(n)$ to be the largest integer $m$ such that $(n!)^m$ divides $2016!$. What is the minimum possible value of $n + k(n)$?
Answer: 89
Problem: A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.
Answer: 217
Problem: Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). How many possible DNA sequences are there on Rthea?
Answer: 28812
Problem: Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_1, y_2, \ldots, y_n$:
\[
(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \ldots = (x+n)^2 + y_n^2.
\]
Answer: 3
Problem: Determine all non-negative integers $k$ such that there exists a function $f : \mathbb{N} \to \mathbb{N}$ satisfying the equation:
\[
f^n(n) = n + k
\]
for all $n \in \mathbb{N}$.
Answer: 0
Problem: Find the sum of all integers $x$, where $x \ge 3$, such that $201020112012_x$ (interpreted as a base $x$ number) is divisible by $x-1$.
Answer: 32
Problem: For how many triples of positive integers $(a, b, c)$ with $1 \leq a, b, c \leq 5$ is the quantity $(a+b)(a+c)(b+c)$ not divisible by $4$?
Answer: 48
Problem: Let $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \(5^m + 63n + 49 = a^k\) holds. Find the minimum value of $k$.
Answer: 5
Problem: The function $f(x) = ax^2 + bx + c$ satisfies the following conditions:
1. $f(\sqrt{2}) = 3$
2. $|f(x)| \le 1$ for all $x \in [-1, 1]$.

Evaluate the value of $f(\sqrt{2013})$.
Answer: 4025
Problem: The integers $1, 2, \ldots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
Answer: 2014
Problem: The number of integral points on the circle with center $ (199, 0) $ and radius $ 199 $ is ________.
Answer: 4
Problem: Let $x, y,$ and $z$ be nonnegative real numbers with $x+y+z=120$. Compute the largest possible value of the median of the three numbers $2x+y, 2y+z,$ and $2z+x$.
Answer: 160
Problem: How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?
Answer: 33
Problem: Let $\triangle AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF$, respectively, such that $BD$ is parallel to $EF$. Point $C$ is chosen in the interior of $\triangle AEF$ such that quadrilateral $ABCD$ is cyclic. If $BC = 3$ and $CD = 4$, then the ratio of the areas $\frac{[ABCD]}{[AEF]}$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
Answer: 5300