ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Suppose $A \subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b \in A$ (where $a$ and $b$ are allowed to be the same), $a + b + 30k$ is always not the product of two consecutive integers. Please find $A$ with the largest possible cardinality.
Answer: 10
Problem: Compute the minimum value of
\[ \frac{x^4 + 2x^3 + 3x^2 + 2x + 10}{x^2 + x + 1} \]
where \( x \) can be any real number.
Answer: 6
Problem: On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.
Answer: 43
Problem: The base-7 representation of the number \( n \) is \( \overline{abc}_{(7)} \), and the base-9 representation of the number \( n \) is \( \overline{cba}_{(9)} \). What is the decimal (base-10) representation of \( n \)?
Answer: 248
Problem: For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary. For example, $d(13) = 1 + 1 + 0 + 1 = 3$. Let
\[
S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3.
\]
Determine $S$ modulo $2020$.
Answer: 1990
Problem: Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.
Answer: 11
Problem: Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[ P(P(x)) - P(x)^2 = x^2 + x + 2016 \] for all real $x$. What is $P(1)$?
Answer: 1010
Problem: Let $n$ be the smallest positive integer such that the remainder of $3n + 45$, when divided by $1060$, is $16$. Find the remainder of $18n + 17$ upon division by $1920$.
Answer: 1043
Problem: Suppose $\overline{a_1a_2\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i = 1, 2, \ldots, 2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$.

(Note: $\overline{xyz\ldots}$ denotes an integer whose digits are $x, y, z, \ldots$.)
Answer: 6
Problem: One day while Tony plays in the backyard of the Kubik's home, he wonders about the width of the backyard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly $12$ feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around $15$ feet. He then measures $2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?
Answer: 82
Problem: Evaluate the following sum:

$$ \frac{1}{\log_2\left(\frac{1}{7}\right)} + \frac{1}{\log_3\left(\frac{1}{7}\right)} + \frac{1}{\log_4\left(\frac{1}{7}\right)} + \frac{1}{\log_5\left(\frac{1}{7}\right)} + \frac{1}{\log_6\left(\frac{1}{7}\right)} - \frac{1}{\log_7\left(\frac{1}{7}\right)} - \frac{1}{\log_8\left(\frac{1}{7}\right)} - \frac{1}{\log_9\left(\frac{1}{7}\right)} - \frac{1}{\log_{10}\left(\frac{1}{7}\right)} $$
Answer: 1
Problem: Let $a$ and $b$ be two positive integers satisfying the equation:
\[ 20\sqrt{12} = a\sqrt{b}. \]
Compute the sum of all possible distinct products $ab$.
Answer: 10800
Problem: Find the smallest positive integer $n$ such that $107n$ has the same last two digits as $n$.
Answer: 50
Problem: The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle DAC$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?
Answer: 90
Problem: The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0$, $P_{1}(x)=x$, and 
\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\]
For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
Answer: 0
Problem: In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.
Answer: 30
Problem: The remainder when $x^{100} - x^{99} + \cdots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a + b$.
Answer: -49
Problem: Numbers $1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2001}$ are written on a blackboard. A student erases two numbers $x, y$ and writes down the number $x + y + xy$ instead. Determine the number that will be written on the board after $2000$ such operations.
Answer: 2001
Problem: Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits.
Answer: 54
Problem: Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.
Answer: 1899
Problem: For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
Answer: 1009375
Problem: Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v \times w \times x \times y \times z$. What is the sum of the integers?
Answer: 5
Problem: Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
Answer: 2592
Problem: For a positive integer $n$, let $A(n)$ be the remainder when $n$ is divided by $11$. Define $T(n) = A(1) + A(2) + A(3) + \ldots + A(n)$. Find the value of $A(T(2021))$.
Answer: 3
Problem: Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such that $A$ and the square are on opposite sides of $BC$. Let the angle bisector of $\angle BAC$ intersect the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.
Answer: 338
Problem: Let $k \ll n$ denote that $k < n$ and $k \mid n$. Let $f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\}$ be such that, if $n \leq 2013$ and $k \ll n$, then $f(k) \ll f(n)$. What is the least possible value of $M$?
Answer: 1024
Problem: A natural number $k > 1$ is called \textit{good} if there exist natural numbers \(a_1 < a_2 < \cdots < a_k\) such that \(\frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1\).

Let \(f(n)\) be the sum of the first \(n\) \textit{good} numbers, \(n \geq 1\). Find the sum of all values of \(n\) for which \(\frac{f(n+5)}{f(n)}\) is an integer.
Answer: 18
Problem: Consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $x, y$, the equation $f(x f(y)) = y f(x)$ holds. Find the smallest possible value of $f(2007)$.
Answer: 18
Problem: Each positive integer $n \geq 1$ is assigned the number $p_n$, which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_{32} = 6$, $p_{203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_{999}$. Find the largest prime that divides $S$.
Answer: 103
Problem: Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?
Answer: 280
Problem: For every pair of real numbers $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$:
\[
   x_n = (1 - a) x_{n - 1} + a y_{n - 1},
   \\ y_n = (1 - b) x_{n - 1} + b y_{n - 1}.
\]
The supermean of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying:
\[
\left (p - \frac{1}{2} \right)^2 + \left (q - \frac{1}{2} \right)^2 \le \left(\frac{1}{10}\right)^2,
\]
the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 307
Problem: A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference between the largest and smallest **CIT Beavers** (over all **Beaver-pairs**).
Answer: 79200
Problem: Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.
Answer: 35
Problem: The centers of the three circles \(A\), \(B\), and \(C\) are collinear with the center of circle \(B\) lying between the centers of circles \(A\) and \(C\). Circles \(A\) and \(C\) are both externally tangent to circle \(B\), and the three circles share a common tangent line. Given that circle \(A\) has radius \(12\) and circle \(B\) has radius \(42\), find the radius of circle \(C\).
Answer: 147
Problem: Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.
Answer: 11621
Problem: How many integers between $2$ and $100$ inclusive cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$?
Answer: 35
Problem: A group of $25$ friends were discussing a large positive integer. The first friend said, "It can be divided by $1$." The second friend said, "It can be divided by $2$." The third friend said, "And by $3$." The fourth friend added, "And by $4$." This pattern continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?
Answer: 787386600
Problem: An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 107
Problem: Define the infinite products:  
\[ A = \prod_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \]  
If \(\frac{A}{B} = \frac{m}{n}\) where \(m,n\) are relatively prime positive integers, determine \(100m+n\).
Answer: 103
Problem: Let $a, b, c$ be real numbers satisfying:
\[ 3ab + 2 = 6b, \quad 3bc + 2 = 5c, \quad 3ca + 2 = 4a. \]
Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s} \in \mathbb{Q}$ and $\frac{t}{u} \in \mathbb{Q}$, both in their lowest terms, find $r + s + t + u$.
Answer: 18
Problem: Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?
Answer: 96
Problem: Show that the number \( 4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right) \) is an integer and determine it.
Answer: 1
Problem: Suppose $a$ and $b$ are positive integers for which $8a^a b^b = 27a^b b^a$. Find $a^2 + b^2$.
Answer: 117
Problem: Determine the value of $q(q(q(2000^{2000})))$, where $q(n)$ represents the sum of the digits of a natural number $n$.
Answer: 4
Problem: There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
Answer: 61
Problem: Given that nonzero real numbers $a, b, c, d$ satisfy $a^b = c^d$ and $\frac{a}{2c} = \frac{b}{d} = 2$, compute $\frac{1}{c}$.
Answer: 16
Problem: Let $N = 30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in \{1, 2, \ldots, N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3 + Bn^2 + 2Cn + D$ is divisible by $N$.
Answer: 2
Problem: The diagram shows a circle with radius $24$ that contains two smaller circles, each with a radius of $12$. These two circles are tangent to each other and to the larger circle. There is also a smallest circle that is tangent to the three other circles. Determine the radius of this smallest circle.
Answer: 8
Problem: **Problem 1.** The least common multiple of the numbers $a$ and $b$ is $88$. If $a^2 + b^2 = 2000$, find the value of $(a + b)$.
Answer: 52
Problem: Given a strictly increasing sequence $a_1 < a_2 < \dots$ of positive integers such that each term $a_k$ is divisible either by 1005 or 1006, but neither term is divisible by 97. Find the least possible value of the maximal difference of consecutive terms $a_{i+1} - a_i$.
Answer: 2010
Problem: Compute the number of positive integer divisors of $2121$ with a units digit of $1$.
Answer: 4
Problem: Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
Answer: 450
Problem: Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes the last stick. For $1 \leq k \leq 100$, determine the number of cases in which the first player can guarantee a win.
Answer: 71
Problem: In an isosceles triangle $ABC$ with an angle $\angle A = 20^\circ$ and base $BC = 12$, a point $E$ on the side $AC$ is chosen such that $\angle ABE = 30^\circ$. Additionally, a point $F$ is on the side $AB$ such that $EF = FC$. Find the length of $FC$.
Answer: 12
Problem: Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probabilities. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.
Answer: 28
Problem: A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle. Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$.
Answer: 4
Problem: Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius $1$. Compute $\frac{120A}{\pi}$.
Answer: 15
Problem: Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.
Answer: 36
Problem: Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
Answer: 28
Problem: Triangle $ABC$ has side lengths $AC=3$, $BC=4$, and $AB=5$. Let $R$ be a point on the incircle $\omega$ of $\triangle ABC$. The altitude from $C$ to $\overline{AB}$ intersects $\omega$ at points $P$ and $Q$. Then, the greatest possible area of $\triangle PQR$ is $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
Answer: 43
Problem: The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:
\[
\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).
\]
Answer: 671
Problem: In a triangle $ABC$, $\angle C = 2\angle A$. A point $D$ is marked on the side $AC$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $BC$?
Answer: 7
Problem: In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$, and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\angle PDF$ in degrees?
Answer: 70
Problem: Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
Answer: 11
Problem: Let $n \geq 3$ be an integer. Mario draws $20$ lines in the plane, such that no two lines are parallel.

For each equilateral triangle formed by three of these lines, Mario receives three coins. For each isosceles and non-equilateral triangle formed by three of these lines, Mario receives one coin. What is the maximum number of coins that Mario can receive?
Answer: 380
Problem: Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers.
Answer: 79
Problem: A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have?
Answer: 13689
Problem: Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt{3} \cdot PS$ and the area of $PQRS$ is $28\sqrt{3}$, what is the length of $PC$?
Answer: 14
Problem: \( \triangle ABC \) has \( AB = 5 \), \( BC = 12 \), and \( AC = 13 \). A circle is inscribed in \( \triangle ABC \), and \( MN \) tangent to the circle is drawn such that \( M \) is on \( \overline{AC} \), \( N \) is on \( \overline{BC} \), and \( \overline{MN} \parallel \overline{AB} \). The area of \( \triangle MNC \) is \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \).
Answer: 43
Problem: Find the number of ordered tuples \((C,A,M,B)\) of non-negative integers such that \[ C! + C! + A! + M! = B! \]
Answer: 7
Problem: Let $a, b, c > 1$ be positive real numbers such that $a^{\log_b c} = 27$, $b^{\log_c a} = 81$, and $c^{\log_a b} = 243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x} + \sqrt{y} + \sqrt{z}$ for positive integers $x, y,$ and $z$. Find $x + y + z$.
Answer: 47
Problem: Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 3
Problem: Given that $a$, $b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, determine the minimum possible positive value of $\frac{1}{a} - \frac{1}{b} - \frac{1}{c}$. This value can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 2101
Problem: Find all positive integers $n$ for which $1^3 + 2^3 + \cdots + 16^3 + 17^n$ is a perfect square.
Answer: 3
Problem: Sohom constructs a square $BERK$ with side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute the area of $CALI$.
Answer: 180
Problem: For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[ s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k). \]
Answer: 9999
Problem: Two circles $S_1$ and $S_2$, with radii $6$ units and $3$ units respectively, are tangent to each other externally. Let $AC$ and $BD$ be their direct common tangents, with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest integer.
Answer: 68
Problem: A particle is at the origin of the Cartesian plane, $(0, 0)$. In each step, the particle can move $1$ unit in any of the four directions: left, right, up, or down. Determine the number of ways the particle can move from $(0, 0)$ to $(0, 2)$ in exactly $6$ steps. 

Note: Two paths are considered different if the order of traversal of each point differs, even if the same set of points is traversed.
Answer: 225
Problem: A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$. The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.
Answer: 20
Problem: Every one of the six trucks of a construction company drove for $8$ hours, and they all together spent $720$ litres of oil. How many litres should $9$ trucks spend if every one of them drives for $6$ hours?
Answer: 810
Problem: Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively.

Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$.
Answer: 2021
Problem: \textbf{Problem 1.} There are less than $400$ marbles. If they are distributed among $3$ children, there is one left over. If they are distributed among $7$ children, there are 2 left over. Finally, if they are distributed among $5$ children, there are none left over. What is the largest number of marbles you have?
Answer: 310
Problem: Emily draws six dots on a piece of paper such that no three lie on a straight line. She then draws a line segment connecting each pair of dots and colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points, there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?
Answer: 1773
Problem: Given a Pacman of radius $1$, and a mouth opening angle of $90^\circ$, what is the largest circular pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.
Answer: 2