ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be $\frac{m}{n}$ where $m,n$ are numbers such that $m,n$ are positive integers and $\gcd(m,n) = 1$. Take this answer and add the numerator and denominator.
Answer: 3004
Problem: Let $ABCDEF$ be a regular hexagon with $AB = 7$. Let $M$ be the midpoint of $DE$. The lines $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, and $AM$ and $BF$ intersect at $R$. Find the value of $[APB] + [BQC] + [ARF] - [PQMR]$, where $[X]$ denotes the area of polygon $X$.
Answer: 0
Problem: Find the last three digits of the number $\frac{2019!}{2^{1009}}$.
Answer: 0
Problem: In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.
Answer: 23
Problem: Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$
Answer: 0
Problem: Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2}, \dots, x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that
\[ n = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}. \]
Answer: 130
Problem: Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
Answer: 11
Problem: At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average, how many minutes can a student spend on a problem?
Answer: 16
Problem: What is the sum of the squares of the roots of the equation $x^2 - 7 \lfloor x \rfloor + 5 = 0$?
Answer: 92
Problem: Triangle $\triangle ABC$ has $AB= 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $\triangle BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m,n$. Find $m+n$.
Answer: 933
Problem: The cells of a $2021 \times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?
Answer: 5
Problem: Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 17
Problem: Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$, satisfying the condition $p \mid (a^{p-2} - a)$?
Answer: 48
Problem: Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2 + xy + y^2 = m$ intersects the hyperbola $xy = n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81 \le n \le 81$.
Answer: 284
Problem: Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation:
\[
LEET + LMT = TOOL
\]
Given that $O = 0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO$.
Answer: 1880
Problem: Find the sum of the real roots of \(2x^4 + 4x^3 + 6x^2 + 4x - 4\).
Answer: -1
Problem: What is the sum of all positive integers less than $30$ that are divisible by $2$, $3$, or $5$?
Answer: 315
Problem: What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?
Answer: 1936
Problem: We draw a triangle inside a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75^\circ$. 

Next, we draw a second triangle congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference, rotated $75^\circ$ clockwise from the first triangle so that it shares a side with the first triangle. 

We continue this process to draw a third, fourth, and fifth triangle, each rotated $75^\circ$ clockwise from the previous triangle. 

The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by this intersection?
Answer: 120
Problem: How many trailing zeros does the value \(300 \cdot 305 \cdot 310 \cdots 1090 \cdot 1095 \cdot 1100\) end with?
Answer: 161
Problem: Compute the number of ordered pairs of integers \((a, b)\), with \(2 \le a, b \le 2021\), that satisfy the equation
\[
a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.
\]
Answer: 43
Problem: Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the following three congruences:

\[
\begin{aligned}
\left\lfloor \frac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\
\left\lfloor \frac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\
\left\lfloor \frac{n}{256} \right\rfloor &\equiv 1 \pmod{4}.
\end{aligned}
\]

Here, $\lfloor \cdot \rfloor$ denotes the greatest integer function.
Answer: 6491
Problem: Let $\text{s}_b(k)$ denote the sum of the digits of $k$ in base $b$. Compute \[ \text{s}_{101}(33) + \text{s}_{101}(66) + \text{s}_{101}(99) + \cdots + \text{s}_{101}(3333). \]
Answer: 6683
Problem: Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive integers.
Answer: 1
Problem: Find the total number of different integers the function

\[ f(x) = \left\lfloor x \right\rfloor + \left\lfloor 2 \cdot x \right\rfloor + \left\lfloor \frac{5 \cdot x}{3} \right\rfloor + \left\lfloor 3 \cdot x \right\rfloor + \left\lfloor 4 \cdot x \right\rfloor \]

takes for $0 \leq x \leq 100.$
Answer: 734
Problem: Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.
Answer: 2189
Problem: Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have:
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$.
Answer: 4
Problem: There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules:  
- If a player holds at least one red card, they pass one red card to the player sitting to their left.  
- Otherwise, they transfer one white card to the left.  

The game ends after the round when each player has one red card and one white card. Determine the maximum number of rounds possible.
Answer: 2009
Problem: In the given figure, the chord $CD$ is perpendicular to the diameter $AB$ and intersects it at point $H$. The length of $AB$ is a two-digit natural number. By changing the order of these two digits, we obtain the length of $CD$. Knowing that the distance from $H$ to the center $O$ is a positive rational number, calculate the length of $AB$.
Answer: 65
Problem: A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$.
Answer: 56
Problem: Find the sum of all integers $n$ that fulfill the equation $$2^n(6-n)=8n.$$
Answer: 9
Problem: Let $R$ be the rectangle on the Cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$.
Answer: 85
Problem: Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$.
Answer: 86400
Problem: In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.
Answer: 3
Problem: Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]

Among all $n$ and partitions of $S$, determine the minimum possible score.
Answer: 10
Problem: The quartic (4th-degree) polynomial $P(x)$ satisfies $P(1) = 0$ and attains its maximum value of $3$ at both $x = 2$ and $x = 3$. Compute $P(5)$.
Answer: -24
Problem: Find the greatest positive integer $N$ with the following property: there exist integers $x_1, \ldots, x_N$ such that $x_i^2 - x_ix_j$ is not divisible by $1111$ for any $i \neq j$.
Answer: 1000
Problem: Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
Answer: 108
Problem: Find the maximum length of a broken line on the surface of a unit cube, such that:
- The line's links are the cube's edges and diagonals of faces.
- The line does not intersect itself.
- The line passes no more than once through any vertex of the cube.
- The endpoints of the line are in two opposite vertices of the cube.
Answer: 7
Problem: Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.
Answer: 540
Problem: Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks. He notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.
Answer: 2517
Problem: In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent to line $BC$ at point $C$. If $r = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $100m + n$.
Answer: 7508
Problem: The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon, Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
Answer: 65
Problem: Let $BC = 6$, $BX = 3$, $CX = 5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX} \perp \overline{BC}$ and $AF = \sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c + 3b$.
Answer: 1288
Problem: At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither.
Answer: 271
Problem: If $x, y, z$ satisfy $x+y+z = 12$, $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$, and $x^3+y^3+z^3 = -480$, find $x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2$.
Answer: 672
Problem: Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits with a black disc. Then Arutyun comes and identifies both closed digits and their order. What is the minimal $N$ for which they can perform such a trick?
Answer: 101
Problem: Let $a$ and $b$ be prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.
Answer: 10000
Problem: Determine all 3-digit numbers which are equal to the cube of the sum of all their digits.
Answer: 512
Problem: Rectangle $ABCD$ has a perimeter of $178$ and an area of $1848$. What is the length of the diagonal of the rectangle?
Answer: 65
Problem: Let $p(x)$ be a polynomial of degree strictly less than $100$ such that it does not have $(x^3-x)$ as a factor. If \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \) for some polynomials $f(x)$ and $g(x)$, then find the smallest possible degree of $f(x)$.
Answer: 0
Problem: In a school, there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.
Answer: 23
Problem: Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S \cup \{15\}$ and $S \cup \{15, 1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?
Answer: 4
Problem: How many ways are there to arrange the numbers $\{ 1,2,3,4,5,6,7,8 \}$ in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.
Answer: 36
Problem: Let $ABC$ be a triangle with $BC = 49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\frac{AB \cdot AC}{-BC + AB + AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
Answer: 250049
Problem: Let $a > b > c > d$ be positive integers satisfying $a + b + c + d = 502$ and $a^2 - b^2 + c^2 - d^2 = 502$. Calculate how many possible values of $a$ are there.
Answer: 124
Problem: Let $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:
1. $x_1^2 + 5x_2^2 = 10$
2. $x_2y_1 - x_1y_2 = 5$
3. $x_1y_1 + 5x_2y_2 = \sqrt{105}$

Find the value of $y_1^2 + 5y_2^2$.
Answer: 23
Problem: A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

1. If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
2. If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}$.
Answer: 2068
Problem: For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.
Answer: 256
Problem: We say that a four-digit number $\overline{abcd}$ is \textit{slippery} if the number $a^4 + b^3 + c^2 + d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ is slippery, since $2^4 + 0^3 + 2^2 + 3 = 23$. How many slippery numbers are there?
Answer: 60
Problem: Rectangle $ABCD$ has side lengths $AB = 6\sqrt{3}$ and $BC = 8\sqrt{3}$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 17
Problem: What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?
Answer: 44
Problem: In trapezoid $ABCD$, the diagonals intersect at $E$. The area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?
Answer: 242
Problem: In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.
Answer: 70
Problem: Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.
Answer: 460
Problem: Compute the remainder when $29^{30} + 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.
Answer: 668
Problem: Find the remainder when $2+4+\cdots+2014$ is divided by $1+3+\cdots+2013$. Justify your answer.
Answer: 1007
Problem: Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
Answer: 20
Problem: Compute the largest prime factor of $357! + 358! + 359! + 360!$.
Answer: 379
Problem: Let $f(x) = (x-a)^3$. If the sum of all $x$ satisfying $f(x) = f(x-a)$ is $42$, find $a$.
Answer: 14
Problem: The lengths of the altitudes of $\triangle ABC$ are the roots of the polynomial $x^3 - 34x^2 + 360x - 1200.$ Find the area of $\triangle ABC.$
Answer: 100
Problem: Given that $a_1, a_2, a_3, \ldots, a_{99}$ is a permutation of $1, 2, 3, \ldots, 99$, find the maximum possible value of
\[
|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.
\]
Answer: 4900
Problem: Find all rational terms of the sequence defined by the formula: $$ a_n = \sqrt{\frac{9n-2}{n+1}}, \quad n \in \mathbb{N} $$
Answer: 3
Problem: Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
Answer: 10
Problem: Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?
Answer: 69
Problem: Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer 
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016$.
Answer: 8
Problem: Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy:
\[
x + y + z = -\frac{\sqrt{3}}{2} - i\sqrt{5}
\]
and
\[
xyz = \sqrt{3} + i\sqrt{5}.
\]
If $x = x_1 + ix_2$, $y = y_1 + iy_2$, and $z = z_1 + iz_2$ for real $x_1, x_2, y_1, y_2, z_1$, and $z_2$, then:
\[
(x_1x_2 + y_1y_2 + z_1z_2)^2
\]
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
Answer: 1516
Problem: A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.
Answer: 62
Problem: What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$?
Answer: 60
Problem: Determine the least real number $k$ such that the inequality
\[
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \geq 4\left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
\]
holds for all real numbers $a, b, c$.
Answer: 8
Problem: Consider the sequence \( \{1, 3, 13, 31, \ldots\} \) that is obtained by following diagonally the array of numbers in a spiral. Find the number in the \(100\)th position of that sequence.
Answer: 39007
Problem: Consider all possible integers $n \ge 0$ such that $5 \cdot 3^m + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.
Answer: 10
Problem: Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$.
Answer: 85
Problem: Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$. What is the maximum possible value of $a + b + c + d + e$?
Answer: 153
Problem: In a plane, 2014 lines are distributed into 3 groups. In each group, all the lines are parallel to each other. What is the maximum number of triangles that can be formed, such that each side of a triangle lies on one of these lines?
Answer: 302561952
Problem: Let $p(x) = 3x^2 + 1$. Compute the largest prime divisor of $p(100) - p(3)$.
Answer: 103
Problem: Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI = 3$ and the distance from $I$ to $BC$ is $2$, what is the square of the length of $BC$?
Answer: 80