ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?
Answer: 10
Problem: Let $f(n) = 9n^5 - 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18), \ldots ,f(2009)$.
Answer: 120
Problem: In $\triangle ABC$, let point $D$ be the foot of the altitude from $A$ to $BC$. Suppose that $\angle A = 90^{\circ}$, $AB - AC = 5$, and $BD - CD = 7$. Find the area of $\triangle ABC$.
Answer: 150
Problem: Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

1. $f(1) = 1.$

2. $\frac{(x + y)}{2} < f(x + y) \le f(x) + f(y)$ for all $x, y \in \mathbb{N}$.

3. $f(4n + 1) < 2f(2n + 1)$ for all $n \ge 0.$

4. $f(4n + 3) \le 2f(2n + 1)$ for all $n \ge 0.$

Find the sum of all possible values of $f(2023)$.
Answer: 1012
Problem: Positive integers $a$, $b$, and $c$ satisfy the equation:
\[
\frac{ab}{a - b} = c
\]
What is the largest possible value of $a + b + c$ that does not exceed 99?
Answer: 99
Problem: Find the number of 5-digit numbers $n$, such that every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$.
Answer: 40
Problem: The function $g(x)$ is defined as $\sqrt{\frac{x}{2}}$ for all positive $x$. 

Calculate the value of the expression:
$$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$

This expression can be expressed as $\cos(b)$ in degrees, where $0^\circ < b < 90^\circ$ and $b = \frac{p}{q}$ for some relatively prime positive integers $p$ and $q$. Find $p+q$.
Answer: 19
Problem: Sokal da is trying to find the largest positive integer $n$ such that when $n$ is transformed to base-7, it appears as twice its value in base-10. For example, $156$ is such a number because $156_{10} = 312_7$ and $312 = 2 \times 156$. Determine Sokal da's number.
Answer: 315
Problem: Point $P$ is taken inside the square $ABCD$ such that $BP + DP = 25$, $CP - AP = 15$, and $\angle ABP = \angle ADP$. What is the radius of the circumcircle of $ABCD$?
Answer: 10
Problem: Call an $8$-digit number \textit{cute} if its digits are a permutation of $1,2,\ldots,8$. For example, $23615478$ is \textit{cute} but $31234587$ is not. Find the number of $8$-digit \textit{cute} numbers that are divisible by $11$.
Answer: 4608
Problem: Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776, -1765, -1754, \ldots$. Joshua computes the (positive) difference between the $1980^\text{th}$ term in the sequence and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?
Answer: 33
Problem: Let $ABCD$ be a square with side length $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, $CD$, and $DA$. A third square $A_2B_2C_2D_2$ is formed in the same way from $A_1B_1C_1D_1$, and a fourth square $A_3B_3C_3D_3$ from $A_2B_2C_2D_2$. Find the sum of the areas of these four squares.
Answer: 120
Problem: Let $\alpha$ and $\beta$ be positive integers such that:
$$
\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .
$$
Find the smallest possible value of $\beta$.
Answer: 23
Problem: Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours.

Mark $k$ squares on the grid. For each unmarked square $x$, there is exactly one marked square $y$ of the same colour such that $x$ and $y$ are connected by a path consisting of squares of the same colour.

Furthermore, for any two marked squares of the same colour, any path connecting them must pass through squares of all the different colours.

Find the maximum value of $k$.
Answer: 2000
Problem: Among the numbers $11n + 10^{10}$, where $1 \le n \le 10^{10}$ is an integer, how many are squares?
Answer: 44801
Problem: Within a group of $2009$ people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends.
Answer: 2006
Problem: Positive integers $x, y, z$ satisfy the equation $x^3 + xy + x^2 + xz + y + z = 301$. Compute $y + z - x$.
Answer: 1
Problem: Find the least positive integer $m$ such that $\text{lcm}(15, m) = \text{lcm}(42, m)$. Here $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$.
Answer: 70
Problem: Consider digits $A, B, C, D$, with $A \neq 0$, such that $ABCD = (CD)^2 - (AB)^2$. Compute the sum of all distinct possible values of $A + B + C + D$.
Answer: 21
Problem: Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that:
\[
Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) 
\]
for some complex number $w$ with $||w||=\frac{\sqrt{3}}{2}$, $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.
Answer: 68
Problem: Find the sum of the prime factors of $67208001$, given that $23$ is one.
Answer: 781
Problem: Given that the polynomial $P(x) = x^5 - x^2 + 1$ has 5 roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.
Answer: 5
Problem: Find the sum of all integers $n$ with $2 \le n \le 999$ that have the following property: if $x$ and $y$ are randomly selected without replacement from the set $\{ 1, 2, \dots, n \}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
Answer: 600
Problem: Find the smallest positive integer $k$ such that 
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] 
for some ordered triple of positive integers $(a,b,c)$.
Answer: 1297
Problem: Let $f$ be a polynomial such that, for all real numbers $x$, \( f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019 \).
Compute \( f(2018) \).
Answer: -2019
Problem: The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 < m < 6.5 \cdot 10^9$. Find $m$.
Answer: 6490716149
Problem: In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?
Answer: 2
Problem: Find the last digit of the number $$\frac{400!}{(200!)(2^{200})}$$
Answer: 5
Problem: You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same direction, meaning their longer sides must be parallel to each other. How many such coverings are possible?
Answer: 257
Problem: Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM = MB = BP = 15$ and $AN = NC = CP = 25$.
Answer: 150
Problem: An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$.  

Let $n!$ denote the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, $24$ is a factorial number since $24 = 4! = 1 \cdot 2 \cdot 3 \cdot 4$.  

Find all perfect numbers greater than $1$ that are also factorials.
Answer: 6
Problem: How many positive integers have a square less than $10^7$?
Answer: 3162
Problem: A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the least number of dominoes that are entirely inside some $2 \times 2$ square?
Answer: 100
Problem: A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube.
Answer: 125
Problem: Let $P(X) = X^5 + 3X^4 - 4X^3 - X^2 - 3X + 4$. Determine the number of monic polynomials $Q(x)$ with integer coefficients such that $\frac{P(X)}{Q(X)}$ is a polynomial with integer coefficients. Note: A monic polynomial is one with leading coefficient $1$ (e.g., $x^3 - 4x + 5$ is monic, but $5x^3 - 4x^2 + 1$ or $x^2 + 3x^3$ are not).
Answer: 12
Problem: In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points.

Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points than any other player. Determine all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.
Answer: 50
Problem: Let $a_1$, $a_2$, \ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence?
Answer: 8
Problem: Is there a whole number that becomes exactly $57$ times less than itself when its first digit is crossed out?
Answer: 7125
Problem: What is the largest prime factor of 8091?
Answer: 31
Problem: Let $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?
Answer: 4544
Problem: Compute the remainder when $98!$ is divided by $101$.
Answer: 50
Problem: Find $\sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z}, m,n \geq 2 \}$.
Answer: 1
Problem: Find the remainder when $7^{7^7}$ is divided by $1000$.
Answer: 343
Problem: Let $a_1, a_2, a_3, \ldots$ be a non-decreasing sequence of positive integers. For $m \ge 1$, define $b_m = \min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n \ge m$. If $a_{19} = 85$, determine the maximum value of $a_1 + a_2 + \cdots + a_{19} + b_1 + b_2 + \cdots + b_{85}$.
Answer: 1700
Problem: How many ordered pairs of integers $(x, y)$ satisfy $2 \cdot |y| \le x \le 40$?
Answer: 841
Problem: For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23} \cdot 23^{20}$, compute
\[\frac{f(N)}{f(f(f(N)))}.\]
Answer: 25
Problem: Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(T_k - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 451
Problem: Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.
Answer: 84
Problem: Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a < b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation:

$$\left[a+\frac{b}{a}\right]=\left[b+\frac{a}{b}\right]$$
Answer: 98
Problem: Find the smallest positive number $\lambda$, such that for any 12 points on the plane $P_1, P_2, \ldots, P_{12}$ (points can overlap), if the distance between any two of them does not exceed 1, then the inequality \( \sum_{1 \le i < j \le 12} |P_iP_j|^2 \le \lambda \) holds.
Answer: 48
Problem: A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.
Answer: 448
Problem: The Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. It is well-known that the sum of any 10 consecutive Fibonacci numbers is divisible by 11. Determine the smallest integer $N$ such that the sum of any $N$ consecutive Fibonacci numbers is divisible by 12.
Answer: 24
Problem: Suppose $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = \cdots = f(x_n) = 2002$ for some distinct integers $x_1, x_2, \ldots, x_n$. Find the largest possible value of $n$.
Answer: 4
Problem: Let $n$ be a positive integer and let $d_1, d_2, \ldots, d_k$ be its divisors, such that $1 = d_1 < d_2 < \ldots < d_k = n$. Find all values of $n$ for which $k \geq 4$ and $n = d_1^2 + d_2^2 + d_3^2 + d_4^2$.
Answer: 130
Problem: Let $S$ be the set of all natural numbers with the property: the sum of the largest three divisors of a number $n$, excluding $n$, is greater than $n$. Determine the largest natural number $k$ that divides every number in $S$.
Answer: 6
Problem: Determine the least positive integer $n$ for which the following statement is true: the product of any $n$ odd consecutive positive integers is divisible by $45$.
Answer: 6
Problem: We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is **kawaii** if \( T \) has the following properties:

1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).

For example, the subset \( T = \{31, 71, 2024\}\) is **kawaii**, but \( T = \{5, 15, 75\}\) is not **kawaii** because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).

What is the largest possible number of elements that a **kawaii** subset can have?
Answer: 675
Problem: The NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16 \times 16$ chess board?
Answer: 252
Problem: Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.
Answer: 66
Problem: If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, find \( m + n \).
Answer: 97
Problem: An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$?
Answer: 9
Problem: Triangle $ABC$ is isosceles with $AB = AC$. The bisectors of angles $\angle ABC$ and $\angle ACB$ meet at point $I$. If the measure of $\angle CIA$ is $130^\circ$, compute the measure of $\angle CAB$.
Answer: 20
Problem: Lines parallel to the sides of an equilateral triangle are drawn such that they divide each side into $10$ equal segments, resulting in the triangle being divided into $100$ congruent smaller triangles. Each of these $100$ triangles is referred to as a "cell." Additionally, lines parallel to each side of the original triangle are drawn through each vertex of the original triangle. The cells located between any two adjacent parallel lines form a "stripe." Determine the maximum number of cells that can be selected such that no two selected cells belong to the same stripe.
Answer: 10
Problem: A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 6425
Problem: The product 
$$\frac{1}{2} \cdot \frac{2}{4} \cdot \frac{3}{8} \cdot \frac{4}{16} \cdot \ldots \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$
is written in its most simplified form. What is the last digit of the denominator?
Answer: 2
Problem: Edward had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. 

The guard fired an arrow with an initial velocity of $100 \frac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \frac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \frac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\frac{\text{ft}}{\text{s}}$)?
Answer: 75
Problem: Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$.
Answer: 783
Problem: Say that a polynomial with real coefficients in two variables, $x, y$, is balanced if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
Answer: 2020050
Problem: If $x + y^{-99} = 3$ and $x + y = -2$, find the sum of all possible values of $x$.
Answer: -195
Problem: 480 one-centimeter unit cubes are used to build a block measuring 6 cm by 8 cm by 10 cm. A tiny ant then chews its way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that it does not "pass through" cubes if it is merely passing through where their edges or vertices meet.
Answer: 20
Problem: Let $\phi = \frac{1+\sqrt{5}}{2}$. A base-$\phi$ number $(a_n a_{n-1} \ldots a_1 a_0)_{\phi}$, where $0 \le a_n, a_{n-1}, \ldots, a_0 \le 1$ are integers, is defined by 
\[
(a_n a_{n-1} \ldots a_1 a_0)_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + \ldots + a_1 \cdot \phi^1 + a_0.
\]
Compute the number of base-$\phi$ numbers $(b_j b_{j-1} \ldots b_1 b_0)_\phi$ which satisfy $b_j \ne 0$ and
\[
(b_j b_{j-1} \ldots b_1 b_0)_\phi = \underbrace{(100 \ldots 100)_\phi}_{\text{Twenty} \ 100's}.
\]
Answer: 1048576
Problem: The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?
Answer: 9
Problem: Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Answer: 716
Problem: There are relatively prime positive integers $p$ and $q$ such that $\frac{p}{q} = \sum_{n=3}^{\infty} \frac{1}{n^5 - 5n^3 + 4n}$. Find $p+q$.
Answer: 97
Problem: Juan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. As a result, Maria's number exceeds Juan's number by $14789$. Find the original number that Juan wrote.
Answer: 532
Problem: In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.
Answer: 4
Problem: Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}$.
Answer: 1983
Problem: A *form* is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, \ldots, m_n$ satisfy $m_1 \geq \ldots \geq m_n$. An *angle* in a *form* consists of a box $v$ and all the boxes to the right of $v$ and all the boxes above $v$. The size of a *form* of an *angle* is the number of boxes it contains. Find the maximum number of *angles* of size $11$ in a form of size $400$.
Answer: 36
Problem: Alex launches his boat into a river and heads upstream at a constant speed. At the same time, at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is flowing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 52
Problem: Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
Answer: 35
Problem: If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
Answer: 27
Problem: How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$?
Answer: 99
Problem: For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
Answer: 611
Problem: Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.
Answer: 730
Problem: Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$ 

For each blank, he flips a fair coin: 
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations. 

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
Answer: 39
Problem: Determine the smallest positive integer $n$ such that there exist positive integers $x_1, x_2, \ldots, x_n$ so that each natural number from 1001 to 2021 inclusive can be expressed as the sum of one or more distinct terms $x_i$ (where $i = 1, 2, \ldots, n$).
Answer: 11
Problem: Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.
Answer: 1923