meetforever/GRPO_CL huggingface dataset:
user: Emily Thorne is hosting a Memorial Day Party to kick off the summer in the Hamptons. She needs to figure out the seating arrangement for her guests. Here are the conditions:

- If she seats $4$ guests per table, there is $1$ guest left over.
- If she seats $5$ guests per table, there are $3$ guests left over.
- If she seats $6$ guests per table, there is $1$ guest left over.

Given that the number of guests is at least $100$ but no more than $200$, what is the greatest possible number of guests?
Answer: 193
user: A $10$-digit number is called interesting if its digits are distinct and the number is divisible by $11111$. Determine the total number of interesting numbers.
Answer: 3456
user: Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal{S}$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal{S}$ such that $OP = 12$. Compute the area of triangle $CDP$.
Answer: 136
user: Two concentric circles have radii $1$ and $4$. Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$. The radius of the six congruent circles can be written $\frac{k+\sqrt{m}}{n}$, where $k, m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$.
Answer: 126
user: Let $\mathbb{Z}^{\geq 0}$ be the set of all non-negative integers. Consider a function $f: \mathbb{Z}^{\geq 0} \to \mathbb{Z}^{\geq 0}$ such that $f(0) = 1$ and $f(1) = 1$, and for any integer $n \geq 1$, we have:
\[
f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.
\]
Determine the value of $\frac{f(2023)}{f(2022)}$.
Answer: 2045254
user: Find the sum of all positive integers less than $2006$ which satisfy the following conditions:
1. The number is a multiple of six.
2. The number is one more than a multiple of seven.
Answer: 47094
user: Let $x, y, z, w$ be integers such that $2^x + 2^y + 2^z + 2^w = 24.375$. Find the value of $xyzw$.
Answer: 72
user: Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$
Answer: 5
user: The odometer of a family car shows \(15,951\) miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting \(15,951\)) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?
Answer: 62
user: Determine the number of integers $2 \le n \le 2016$ such that $n^n - 1$ is divisible by $2$, $3$, $5$, and $7$.
Answer: 9
user: Determine the number of sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$, the number $\frac{x_{m+1}}{x_m}$ is a prime number.
Answer: 252
user: Pratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, while Payel multiplies his number by $2$ and then adds $2$ to it. In other words, on the first day, their numbers will be $(2n - 2)$ and $(2m + 2)$ respectively. Find the minimum integer $x$ with proof such that if $n - m \geq x$, then Pratyya's number will be larger than Payel's number every day.
Answer: 4
user: Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
Answer: 11
user: We have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
Answer: 12
user: If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\triangle CDX$ is equilateral, compute $\angle AXE$ in degrees.
Answer: 84
user: Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0, 1, 2, 3$ occurs at least once in them.
Answer: 490
user: In a circus, there are $n$ clowns who dress and paint themselves using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.
Answer: 48
user: How many four-digit positive integers have exactly one digit equal to $1$ and exactly one digit equal to $3$?
Answer: 720
user: In a quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, and $BD = 24$. If $r$ is the radius of the circle inscribable in the quadrilateral, what is the integer closest to $r$?
Answer: 8
user: Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square, it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2 \times 4$ rectangle instead of $2 \times 3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C_1, C_2, \ldots, C_n$ which are all distinct such that the knight starts at the $C_1$ square and for each $i$ from $1$ to $n-1$, it can use the movement described before to go from the $C_i$ square to the $C_{i+1}$.

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5 \times 5$ board.
Answer: 12
user: Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?

```asy
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
```
Answer: 540
user: Let \( \triangle ABC \) be a triangle with \( \angle A = 90^\circ \). The bisectors of angles \( B \) and \( C \) intersect at point \( I \). Suppose that \( IH \) is perpendicular to \( BC \) and \( H \) lies on \( BC \). If \( HB = 5 \text{ cm} \) and \( HC = 8 \text{ cm} \), compute the area of \( \triangle ABC \).
Answer: 40
user: Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
Answer: 5
user: Find the number of even integers \( n \) such that \( 0 \le n \le 100 \) and \( 5 \mid n^2 \cdot 2^{2n^2} + 1 \).
Answer: 20
user: Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(ab + cd = 10\).
Answer: 58
user: Determine the last four decimal digits of the number $1997 \cdot 5^{1998}$.
Answer: 3125
user: Find the highest power of $2$ that divides exactly into $1996! = 1 \times 2 \times \cdots \times 1996$.
Answer: 1989
user: The set $A=\{1,2,3,\cdots, 10\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if it contains $n$ as an element. A subset of $A$ is minimally competent if it itself is competent, but none of its proper subsets are. Find the total number of minimally competent subsets of $A$.
Answer: 55
user: Let there be 50 natural numbers $a_i$ such that $0 < a_1 < a_2 < \ldots < a_{50} < 150$. What is the greatest possible sum of the differences $d_j$ where each $d_j = a_{j+1} - a_j$?
Answer: 148
user: Define a finite sequence $\left( s_i \right)_{1\le i\le 2004}$ with initial conditions $s_0 + 2 = s_1 + 1 = s_2 = 2$ and the recurrence relation:
$$ s_n = 1 + s_{n-1} + s_{n-2} - s_{n-3}. $$
Calculate its last element.
Answer: 1005006
user: Compute the largest positive integer $n$ such that $\frac{2007!}{2007^n}$ is an integer.
Answer: 9
user: For $n$ a positive integer, denote by $P(n)$ the product of all positive integer divisors of $n$. Find the smallest $n$ for which 

\[ P(P(P(n))) > 10^{12} \]
Answer: 6
user: A contest consists of six problems, each worth 7 points. For any given problem, a contestant can score either $0$, $1$, or $7$ points. Determine the number of possible total scores a contestant can achieve across all six problems.
Answer: 28
user: Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ \perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}$. Find $\angle CBA$, in degrees.
Answer: 90
user: Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}(\omega^{a_p} - \omega^{a_q}).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.
Answer: 1054
user: The side lengths of a triangle are natural number multiples of $7$, and each side is smaller than $40$. How many triangles satisfy these conditions?
Answer: 22
user: For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^\circ$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^\circ$ or $60^\circ$, and the triangle should definitely not be isosceles.

How many different triangles can Tina draw? (Similar triangles are considered the same.)
Answer: 0
user: A positive integer is considered **happy** if:

1. All its digits are different and not $0$.
2. One of its digits is equal to the sum of the other digits.

For example, 253 is a **happy** number. How many **happy** numbers are there?
Answer: 264
user: A number is called \textit{6-composite} if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is \textit{composite} if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
Answer: 441
user: Suppose that $20^{21} = 2^a 5^b = 4^c 5^d = 8^e 5^f$ for positive integers $a, b, c, d, e,$ and $f$. Find $\frac{100bdf}{ace}$.
Answer: 75
user: Define the two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations:
- $a_{n+1} = 3a_n + b_n$
- $b_{n+1} = 3b_n - a_n$

for all nonnegative integers $n$. Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31$, respectively. Compute $100r + s$.
Answer: 3010
user: The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. Given that $P(3) = 89$, determine the value of $P(10)$.
Answer: 859
user: Let $f$ be a quadratic function that satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$.

For two distinct real numbers $a$ and $b$, if $f(a) = f(b)$, then $f(a^2 - 6b - 1) = f(b^2 + 8)$.
Answer: 13
user: Determine the sum of the positive integers $n$ such that there exist primes $p, q, r$ satisfying $p^n + q^2 = r^2$.
Answer: 5
user: Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$.
Answer: 1012
user: The four sets $A$, $B$, $C$, and $D$ each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets?
Answer: 1094
user: For a natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \leq n$, define an $n,s,k$-block as a string of $k$ consecutive elements from $s$. We say that two $n,s,k$-blocks, namely, $a_1a_2\ldots a_k$ and $b_1b_2\ldots b_k$, are incompatible if there exists an $i \in \{1,2,\ldots ,k\}$ such that $a_i \neq b_i$. Additionally, for two natural numbers $r \leq n$ and $l$, we say that $s$ is $r,l$-typed if there are at most $l$ pairwise incompatible $n,s,r$-blocks. 

Let $t$ be a $3,7$-typed string consisting of $10,000$ binary digits. Determine the maximum number $M$ such that $t$ is $10,M$-typed.
Answer: 128
user: Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.
Answer: 80
user: Determine all even numbers $n$, $n \in \mathbb{N}$ such that:
\[
\frac{1}{d_{1}} + \frac{1}{d_{2}} + \cdots + \frac{1}{d_{k}} = \frac{1620}{1003},
\]
where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.
Answer: 2006
user: Find all prime numbers of the form $\frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}$, where $n$ is a natural number.
Answer: 101
user: Find the sum of all the positive integers that are divisors of either $96$ or $180$.
Answer: 770
user: Find the largest positive integer $n$ ($n \ge 3$), such that there exists a convex $n$-gon where the tangent of each interior angle is an integer.
Answer: 8
user: Find the smallest positive integer $n$ such that $\sqrt{\frac{1^2 + 2^2 + \ldots + n^2}{n}}$ is an integer.
Answer: 337
user: Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are:

- Two black cards
- Two red cards
- One black and one red card

Determine the probability that Hong and Song play exactly the same pair of cards in each of the four turns. This probability is given as $\frac{m}{n}$, where $m$ and $n$ are positive integers such that $\gcd(m, n) = 1$. Calculate $100m + n$.
Answer: 25150
user: Suppose that $n$ is a positive integer and that $a$ is the integer equal to \( \frac{10^{2n}-1}{3(10^n+1)} \). If the sum of the digits of $a$ is 567, what is the value of $n$?
Answer: 189
user: Compute the number of digits in $2015!$.
Answer: 5786
user: A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$.
Answer: 1257
user: Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a > b > c > d$ are positive integers satisfying:

$$|\zeta^a + \zeta^b + \zeta^c + \zeta^d| = \sqrt{3}.$$  

Compute the smallest possible value of $1000a + 100b + 10c + d$.
Answer: 7521
user: How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer root?
Answer: 49
user: Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions:

- $f(1) = 2$
- $f(a) + f(b) \leq 2\sqrt{f(a)}$
- For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b + 100)$.
Answer: 10201
user: There are $360$ permutations of the letters in $MMATHS$. When ordered alphabetically, starting from $AHMMST$, $MMATHS$ is in the $n$th permutation. What is $n$?
Answer: 173
user: Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a \neq 20, 22$. Find the minimum possible positive value of $P(0)$.
Answer: 20
user: In the triangle $ABC$, $\angle ABC = \angle ACB = 78^\circ$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^\circ$, $\angle CBE = 51^\circ$. Find the measure of angle $\angle BED$.
Answer: 12
user: Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$, where $n$ and $p$ are squarefree, find $m+n+p$.
Answer: 11
user: Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2 + b^2 + c^2 + a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m \rfloor$.
Answer: 594
user: What are the last two digits of the number $$(11 + 12 + 13 + \ldots + 2006)^2$$?
Answer: 56
user: In $\triangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Draw the circumcircle of $\triangle ABC$, and suppose that the circumcircle has center $O$. Extend $AO$ past $O$ to a point $D$, $BO$ past $O$ to a point $E$, and $CO$ past $O$ to a point $F$ such that $D, E, F$ also lie on the circumcircle. Compute the area of the hexagon $AFBDCE$.
Answer: 168
user: Compute the number of sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold:

1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$.
2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}$ such that $a_i + a_j - N$ is divisible by $203$.
Answer: 20503
user: The repeating decimal $2.0151515\ldots$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 199
user: What is the minimal number of operations needed to repaint an entirely white grid $100 \times 100$ to be entirely black, if in one move we can choose $99$ cells from any row or column and change their color?
Answer: 200
user: Compute the number of integer solutions $(x, y)$ to the equation $xy - 18x - 35y = 1890$.
Answer: 96
user: Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?
Answer: 840
user: If $m$, $n$, and $p$ are three different natural numbers, each between $2$ and $9$, what are all the possible integer values of the expression $\frac{m+n+p}{m+n}$?
Answer: 2
user: Pick out three numbers from $0, 1, \ldots, 9$, such that their sum is an even number and not less than $10$. We have ________ different ways to pick these numbers.
Answer: 51
user: What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits is the same when read backwards.)
Answer: 21
user: For each $n \in \mathbb{N}$, let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find the value of $d_1 + d_2 + \cdots + d_{2018} + d_{2019}$.
Answer: 6725
user: Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with every possible height-weight combination.

The USAMTS wants to field a competitive team, so there are some strict requirements:

- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team.

Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?
Answer: 128
user: The intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.
Answer: 25
user: A board measuring $64$ inches in length and $4$ inches in height is inclined such that the longer side of the board forms a $30^\circ$ angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is the value of $a + b + c$?
Answer: 37
user: In the figure below, each side of the rhombus has a length of 5 centimeters. 

\[ \text{[asy]} 
  \text{import graph;} 
  \text{unitsize}(2.0\text{cm}); 
  \text{real } w = \sqrt{3}; 
  \text{draw}((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- \text{cycle}); 
  \text{filldraw}(
    \text{Circle}((-0.5, 0), \frac{0.8}{\sqrt{\pi}}), \text{gray}
  ); 
  \text{label}("$60^\circ$", (w - 0.1, 0), \text{W}); 
\]

The circle lies entirely within the rhombus. The area of the circle is \( n \) square centimeters, where \( n \) is a positive integer. Compute the number of possible values of \( n \).
Answer: 14
user: Find the sum of all real numbers $x$ such that \( x^2 = 5x + 6\sqrt{x} - 3 \).
Answer: 7
user: What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times \ldots \times 50$ is divisible by $10^n$?
Answer: 11
user: We call $\triangle ABC$ with centroid $G$ "balanced" on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B$. $\triangle XYZ$, with $XY=2023$ and $\angle ZXY=120^\circ$, is balanced on $XY$. What is the maximum value of $XZ$?
Answer: 4046
user: Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$, $C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $\triangle AJG$ is an equilateral triangle, the area of $\triangle AJG$ is $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 48
user: A positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$.
Answer: 7
user: Calculate the least integer greater than $5^{(-6)(-5)(-4)...(2)(3)(4)}$.
Answer: 2
user: Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
Answer: 22200
user: Ryan is altering Brice’s coin so that one side appears twice as often as the other. He randomly decides to weight either heads or tails more heavily, with equal probability.

Brice flips this modified coin twice, and it lands on heads both times. We need to find the probability that the coin lands on heads in the next flip. This probability can be expressed as $\frac{p}{q}$, where $p$ and $q$ are positive integers and $\gcd(p, q) = 1$. Determine the value of $p + q$.
Answer: 8
user: Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.
Answer: 420
user: Let $a$ and $b$ be integer solutions to $17a + 6b = 13$. What is the smallest possible positive value for $a - b$?
Answer: 17