ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
Answer: 47
Problem: What is the closest number to $169$ that is divisible by $9$?
Answer: 171
Problem: Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$. A subset $S$ of $T$ is defined as \textit{handy} if the sum of all the elements of $S$ is a multiple of $5$. For example, the empty set is handy (because its sum is $0$) and $T$ itself is handy (because its sum is $120$). Compute the number of handy subsets of $T$.
Answer: 6560
Problem: Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt{3}$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$.
Answer: 243
Problem: Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Define $a_1 = 1997^{1996^{1997}}$, and for every $n > 0$, let $a_{n+1} = \operatorname{cif}(a_n)$. Find $\lim_{n \to \infty} a_n$.
Answer: 1
Problem: There is a complex number $K$ such that the quadratic polynomial $7x^2 + Kx + 12 - 5i$ has exactly one root, where $i = \sqrt{-1}$. Find $|K|^2$.
Answer: 364
Problem: A fair $6$-sided die is repeatedly rolled until a $1$, $4$, $5$, or $6$ is rolled. What is the expected value of the product of all the rolls?
Answer: 16
Problem: Line segment $\overline{AB}$ has a perpendicular bisector $\overline{CD}$, where $C$ is the midpoint of $\overline{AB}$. The segments have lengths $AB = 72$ and $CD = 60$. Let $R$ be the set of points $P$ that are midpoints of line segments $\overline{XY}$, where $X$ lies on $\overline{AB}$ and $Y$ lies on $\overline{CD}$. Find the area of the region $R$.
Answer: 1080
Problem: Let \[ f(x) = \cos(x^3 - 4x^2 + 5x - 2). \]
If we let \( f^{(k)} \) denote the \( k \)th derivative of \( f \), compute \( f^{(10)}(1) \). For the sake of this problem, note that \( 10! = 3628800 \).
Answer: 907200
Problem: Define
\[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 4045
Problem: Let $N = \prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.
Answer: 624
Problem: Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step:
- If the number $n$ on the board is even, he replaces it with $\frac{1}{2}n$.
- If the number $n$ on the board is odd, he replaces it with $n^2 + 3$.

Determine for how many choices of starting numbers below $2023$ Felix will never write a number of more than four digits on the board.
Answer: 21
Problem: Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \dots, X_n$ in the plane with $AB \neq CD$ that satisfy the following condition: for each $i=1,2,\dots,n$, the triangles $\triangle ABX_i$ and $\triangle CDX_i$ are equal.
Answer: 4
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: Determine the smallest positive integer $M$ with the following property: For every choice of integers $a, b, c$, there exists a polynomial $P(x)$ with integer coefficients such that $P(1) = aM$, $P(2) = bM$, and $P(4) = cM$.
Answer: 6
Problem: Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.)
Answer: 128
Problem: Integers $a$, $b$, $c$ are selected independently and at random from the set $\{ 1, 2, \cdots, 10 \}$, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$.
Answer: 136
Problem: Let $a_1, a_2, a_3, \dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n$. Suppose $a_1 = 1$, $a_2 = 3$,
\[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] 
Then $c - b + \frac{c-b}{b}$ can be written in the form $x^y$, where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y$.
Answer: 2024
Problem: How many ways are there to rearrange the letters of $CCAMB$ such that at least one $C$ comes before the $A$?
Answer: 40
Problem: The equation $x^5 - 2x^4 - 1 = 0$ has five complex roots $r_1, r_2, r_3, r_4, r_5$. Find the value of 
\[
\frac{1}{r_1^8} + \frac{1}{r_2^8} + \frac{1}{r_3^8} + \frac{1}{r_4^8} + \frac{1}{r_5^8}.
\]
Answer: 16
Problem: Let $ABCDEF$ be a regular hexagon. Let $P$ be the intersection point of $\overline{AC}$ and $\overline{BD}$. Suppose that the area of triangle $EFP$ is 25. What is the area of the hexagon?
Answer: 90
Problem: Find $$\sum^{k=672}_{k=0} \binom{2018}{3k+2} \pmod{3}$$
Answer: 1
Problem: Let \[ \begin{array}{ccccccccccc} A &=& 5 \cdot 6 &-& 6 \cdot 7 &+& 7 \cdot 8 &-& \cdots &+& 2003 \cdot 2004, \\ B &=& 1 \cdot 10 &-& 2 \cdot 11 &+& 3 \cdot 12 &-& \cdots &+& 1999 \cdot 2008. \end{array} \] 
Find the value of $A-B$.
Answer: 20
Problem: Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\frac{AW}{BW} = \frac{2}{9}$, $\frac{AX}{CX} = \frac{5}{6}$, and $\frac{DY}{EY} = \frac{9}{2}$. The ratio $\frac{EZ}{FZ}$ can then be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 33
Problem: An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers.
Answer: 1360
Problem: Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?
Answer: 503
Problem: All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in the computer's memory. The program looks through all the sequential pairs (first and second, second and third,...) and exchanges numbers in the pair if the number in the lower place is greater than the other. Then the program repeats the process but moves from the other end of the array. The number that initially stands on the $100$-th place reserves its place. Find that number.
Answer: 100
Problem: Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve Daily Challenge problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving Daily Challenge problem $D_i$. In how many satisfactory orders can he complete all six actions?
Answer: 90
Problem: Jackson begins at $1$ on the number line. At each step, he remains in place with a probability of $85\%$ and increases his position on the number line by $1$ with a probability of $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\frac{1}{d_n}$. Find the least $n$ such that $\frac{1}{E_n} > 2017$.
Answer: 13446
Problem: How many ways are there to color the vertices of a cube using the colors red, blue, or green such that no edge connects two vertices of the same color? Note that rotations and reflections are considered distinct colorings.
Answer: 114
Problem: Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?
Answer: 90
Problem: Let circle $O$ have radius $5$ with diameter $\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\overline{FA}$ and $\overline{FE}$ intersect circle $O$ at points $B$ and $D$, respectively. If $FA = 10$ and $m \angle FAE = 30^\circ$, then the perimeter of quadrilateral $ABDE$ can be expressed as $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}$, where $a, b, c$, and $d$ are rational. Find $a + b + c + d$.
Answer: 15
Problem: For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes, respectively. A positive integer $n$ is called fair if $s(n) = c(n) > 1$. Find the number of fair integers less than $100$.
Answer: 7
Problem: Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:

- $M$ is the sum of the squares of some $n$ consecutive positive integers.
- $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
Answer: 4250
Problem: A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct 3-letter words can be made from a subset of the letters \(c, o, m, b, o\), where each letter in the list is used no more than the number of times it appears?
Answer: 33
Problem: What is the least positive integer $n$ such that $9n$ is a perfect square and $12n$ is a perfect cube?
Answer: 144
Problem: Twenty-five of the numbers $1, 2, \ldots, 50$ are chosen. Twenty-five of the numbers $51, 52, \ldots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers.
Answer: 2525
Problem: The 42 points $P_1, P_2, \ldots, P_{42}$ lie on a straight line in that order, such that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1 \leq n \leq 41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
Answer: 861
Problem: A sequence of real numbers $\{a_n\}_{n = 1}^\infty$ with $n=1,2,...$ has the following property:
\[
6a_n + 5a_{n-2} = 20 + 11a_{n-1} \quad (\text{for } n \geq 3).
\]
The first two elements are $a_1=0$ and $a_2=1$. Find the integer closest to $a_{2011}$.
Answer: 40086
Problem: There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 5
Problem: The cubic polynomials $p(x)$ and $q(x)$ satisfy the following conditions:  
1. $p(1) = q(2)$  
2. $p(3) = q(4)$  
3. $p(5) = q(6)$  
4. $p(7) = q(8) + 13$  

Find $p(9) - q(10)$.
Answer: 52
Problem: Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$. He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne makes it rain $n$ days from today is greater than $49.9\%$.
Answer: 9
Problem: How many positive integers less than 2015 have exactly 9 positive integer divisors?
Answer: 13
Problem: For how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in the set of real numbers?
\[
\begin{cases}
  a^2 + b^2 = kc(a+b), \\
  b^2 + c^2 = ka(b+c), \\
  c^2 + a^2 = kb(c+a).
\end{cases}
\]
Answer: 2
Problem: Let $a$ be a positive real number such that $\frac{a^2}{a^4-a^2+1}=\frac{4}{37}$. Then $\frac{a^3}{a^6-a^3+1}=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 259
Problem: Find the remainder when \( \prod_{n=3}^{33} \left( 2n^4 - 25n^3 + 33n^2 \right) \) is divided by \( 2019 \).
Answer: 0
Problem: Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was $47$. If Andrea is $29$ years older than Jim, what is the sum of their ages now?
Answer: 79
Problem: How many positive integers $a$ with $a \le 154$ are there such that the coefficient of $x^a$ in the expansion of 
\[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] 
is zero?
Answer: 60
Problem: On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves.
Answer: 67
Problem: Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
Answer: 4
Problem: Suppose there exist constants $A$, $B$, $C$, and $D$ such that 
\[n^4 = A\binom{n}{4} + B\binom{n}{3} + C\binom{n}{2} + D\binom{n}{1}\]
 holds true for all positive integers $n \geq 4$. What is $A + B + C + D$?
Answer: 75
Problem: An event occurs periodically over $x$ consecutive years, followed by a break of $y$ consecutive years. The event occurred in the years 1964, 1986, 1996, and 2008, and did not occur in 1976, 1993, 2006, or 2013. Determine the first year in which the event will occur again.
Answer: 2018
Problem: Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).
Answer: 19
Problem: Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?
Answer: 5
Problem: Let $a$, $b$, and $c$ be the 3 roots of $x^3 - x + 1 = 0$. Find $\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1}$.
Answer: -2
Problem: Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.
Answer: 6
Problem: Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?
Answer: 640
Problem: By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?
Answer: 12
Problem: How many ways are there to fill in a $3 \times 3$ grid of cells with $0$'s and $2$'s, one number in each cell, such that each $2 \times 2$ contiguous subgrid contains exactly three $2$'s and one $0$?
Answer: 8
Problem: There is a right triangle $\triangle ABC$ where $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ such that $\angle ACX = \angle XCY = \angle YCZ = \angle ZCB$ and $BZ = 2AX$. The smallest angle of $\triangle ABC$ is $\frac{a}{b}$ degrees, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. Find $a + b$.
Answer: 277
Problem: Let $ABC$ be a triangle with $\angle A = 60^\circ$. Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to $\ell$. Given that $AB = 20$ and $AC = 22$, compute $XY$.
Answer: 21
Problem: Given positive real numbers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2$ and $ab = c$. Determine the value of the expression:
\[
\left| \frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2} \right|
\]
Answer: 4
Problem: A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked $5$ times as many hours as the last woman, for how many hours did the first woman work?
Answer: 75
Problem: In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$. Given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.
Answer: 260
Problem: Among $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$.
Answer: 1037
Problem: PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, what is the probability that the 4th customer orders a previously unordered item? Express the probability as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 189
Problem: Evaluate the following expression:

\[
\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}
\]
Answer: 2005
Problem: What is the greatest common factor of $12345678987654321$ and $12345654321$?
Answer: 12321
Problem: A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the sum of the largest and smallest possible value of $x$.
Answer: 120
Problem: In triangle $ABC$, $AB = 8$, $AC = 9$, and $BC = 10$. Let $M$ be the midpoint of $BC$. Circle $\omega_1$ with area $A_1$ passes through $A$, $B$, and $C$. Circle $\omega_2$ with area $A_2$ passes through $A$, $B$, and $M$. Then $\frac{A_1}{A_2} = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 16295
Problem: The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.
Answer: 60
Problem: For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \leq p(n)$.
Answer: 13
Problem: Let $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of 
\[ \sum_{i=1}^{5} |a_i - i| = |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + |a_4 - 4| + |a_5 - 5|. \]
Answer: 8
Problem: If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.
Answer: 15
Problem: Calculate $1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 + 9 + \cdots - 96 + 97 + 98 + 99 + 100$.
Answer: 202
Problem: It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam?
Answer: 10
Problem: Find the smallest positive integer $n$ such that if $n = p_1^{a_1} p_2^{a_2} \ldots p_s^{a_s}$, then \[ \phi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \ldots \left(1 - \frac{1}{p_s}\right) = \frac{2^5}{47}n. \]
Answer: 59455
Problem: How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x \leq 29$, a subset of $A$ sums to $x$?
Answer: 4
Problem: Six chairs are arranged in a row, and six people randomly choose a seat among these chairs. Each person then decides one of three positions for their legs:  
1. Feet on the floor  
2. Legs crossed to the right  
3. Legs crossed to the left  

A problem arises only if two adjacent people have the following leg positions:  
- The person on the right crosses their legs to the left.  
- The person on the left crosses their legs to the right.

Determine the probability that this situation does **not** occur. Express the probability as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the sum $m+n$.
Answer: 1106
Problem: Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that:
- $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$;
- For any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$.

Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions.
Answer: 7993
Problem: Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ  ... \circ  \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = \min \{k \in \mathbb{N} \mid \phi^k (n)=1 \}$.

For example:
- $\phi^1 (13) = \phi(13) = 12$
- $\phi^2 (13) = \phi (\phi (13)) = 4$
- $\phi^3 (13) = \phi(\phi(\phi(13))) = 2$
- $\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$

Thus, $\theta (13) = 4$.

Let $f(r) = \theta (13^r)$. Determine $f(2012)$.
Answer: 6037
Problem: Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(n) + f(m)) = m + n$ for all positive integers $n, m$. Find the possible value for $f(1988)$.
Answer: 1988
Problem: 100 schoolchildren participated in the Mathematical Olympiad where 4 tasks were given. The first problem was solved by 90 students, the second by 80, the third by 70, and the fourth by 60. No student solved all the problems. Students who solved both the third and fourth questions received an award. How many students were awarded?
Answer: 30
Problem: The unit price of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed using the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to make only one purchase, what is the maximum possible number of screws he can buy?
Answer: 252
Problem: A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$?
Answer: 1347
Problem: What are the last two digits of $2003^{2005^{2007^{2009}}}$, where $a^{b^c}$ means $a^{(b^c)}$?
Answer: 43
Problem: The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.
Answer: 666
Problem: Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1 \pmod{2017}$.
Answer: 30576
Problem: Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \leq n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.
Answer: 540
Problem: If $x$, $y$, $z$ are real numbers satisfying:
\[
\begin{align*}
(x + 1)(y + 1)(z + 1) &= 3, \\
(x + 2)(y + 2)(z + 2) &= -2, \\
(x + 3)(y + 3)(z + 3) &= -1,
\end{align*}
\]
find the value of
\[ (x + 20)(y + 20)(z + 20). \]
Answer: 6748