meetforever/GRPO_CL huggingface dataset:
user: In how many ways can we change the sign $*$ with $+$ or $-$, such that the following equation is true?
\[ 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 29 \]
Answer: 8
user: Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1 < k < n$ and $\gcd(k, n) = 1$ is a prime.
Answer: 105
user: Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$, and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A$, $y \in B$, $z \in C$, and $x + y + z = 36$.
Answer: 46
user: Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 101
user: Find the sum of the x-coordinates of the distinct points of intersection of the plane curves given by $x^2 = x + y + 4$ and $y^2 = y - 15x + 36$.
Answer: 0
user: Find all prime numbers $p$ such that there exist positive integers $x$ and $y$ satisfying the following system of equations:
\[
\begin{cases}
  p + 49 = 2x^2 \\
  p^2 + 49 = 2y^2
\end{cases}
\]
Answer: 23
user: Tony's favorite "sport" is a spectator event known as the $\textit{Super Mega Ultra Galactic Thumbwrestling Championship}$ (SMUG TWC). During the $2008$ SMUG TWC, $2008$ professional thumb-wrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of $\textit{Thumbzilla}$. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOX$^\text{TM}$ television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let $m$ be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute $m$.
Answer: 1007012
user: Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30\% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. 

Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs, $T$. Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then, $T$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime. Find $m+n$.
Answer: 101
user: Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?
Answer: 23
user: Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[(ax+by)^3 + (cx+dy)^3 = x^3 + y^3\] holds for all complex numbers $x, y$.
Answer: 18
user: In the diagram below, rectangle $ABCD$ has $AB = 5$ and $AD = 12$. Also, $E$ is a point in the same plane outside $ABCD$ such that the perpendicular distances from $E$ to the lines $AB$ and $AD$ are $12$ and $1$, respectively, and $\triangle ABE$ is acute. There exists a line passing through $E$ which splits $ABCD$ into two figures of equal area. Suppose that this line intersects $\overline{AB}$ at a point $F$ and $\overline{CD}$ at a point $G$. Find $FG^2$.
Answer: 145
user: Every cell of a $3 \times 3$ board is colored either red or blue. Find the number of all possible colorings such that there are no $2 \times 2$ squares where all cells are red.
Answer: 417
user: Let us consider a polynomial $P(x)$ with integer coefficients satisfying:
\[
P(-1) = -4, \\
P(-3) = -40, \\
P(-5) = -156.
\]
What is the largest possible number of integers $x$ satisfying:
\[
P(P(x)) = x^2?
\]
Answer: 0
user: Let $A$ be a set of numbers chosen from $1, 2, \ldots, 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non-equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?
Answer: 10
user: Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.
Answer: 37
user: Sam Wang decides to evaluate an expression of the form $x + 2 \cdot 2 + y$. However, he unfortunately reads each 'plus' as a 'times' and reads each 'times' as a 'plus'. Surprisingly, he still gets the problem correct. Find $x + y$.
Answer: 4
user: Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$ Find $n$ such that the first $n$ terms sum up to $2010.$
Answer: 1027
user: The numbers $a, b, c, d$ are $1, 2, 2, 3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$?
Answer: 512
user: In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos^2 \angle AEI = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, determine $m+n$.
Answer: 55
user: Determine the least $n \in \mathbb{N}$ such that $n! = 1 \cdot 2 \cdot 3 \cdots (n-1) \cdot n$ has at least 2010 positive factors.
Answer: 14
user: Find the sum of all positive integers $n$ such that $\tau(n)^2 = 2n$, where $\tau(n)$ is the number of positive integers dividing $n$.
Answer: 100
user: The picture below shows how Juan wants to divide a square field into three regions, all sharing a well at vertex $B$. If the side length of the field is $60$ meters, and each of the three regions has the same area, how far must the points $M$ and $N$ be from $D$? Note: The area of each region includes the area occupied by the well.

```latex
\begin{asy}
pair A=(0,0), B=(60,0), C=(60,-60), D=(0,-60), M=(0,-40), N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));
\end{asy}
```

Unrelated information has been removed for clarity.
Answer: 20
user: Ana, Beatriz, Carlos, Diego, and Emilia play a chess tournament. Each player faces each of the other four only once. Each player gets $2$ points if they win the match, $1$ point if they draw, and $0$ points if they lose. At the end of the tournament, it turns out that the scores of the $5$ players are all different. Find the maximum number of ties there could be in the tournament and justify why there could not be a higher number of ties.
Answer: 4
user: For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. For example, if $n = 260$, then $n' = 26$, and if $n = 2020$, then $n' = 22$. Find the number of three-digit numbers $n$ for which $n'$ is a divisor of $n$ and is different from $n$.
Answer: 93
user: Let $ABCD$ be a square with side length $6$. Circles $X$, $Y$, and $Z$ are congruent circles with centers inside the square such that:

- Circle $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$.
- Circle $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$.
- Circle $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$.

The radius of circle $X$ can be expressed as $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 195
user: Suppose rectangle $FOLK$ and square $LORE$ are on the plane such that $RL = 12$ and $RK = 11$. Compute the product of all possible areas of triangle $RKL$.
Answer: 414
user: Let $X = (x_1, x_2, \ldots, x_9)$ be a permutation of the set $\{1, 2, \ldots, 9\}$. Let $A$ be the set of all such permutations $X$. For any $X \in A$, define the function $f(X) = x_1 + 2x_2 + \cdots + 9x_9$. Let $M = \{f(X) \mid X \in A\}$. Find $|M|$, where $|S|$ denotes the number of elements in the set $S$.
Answer: 121
user: Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{\text{max}}$ and $P_{\text{min}}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{\text{max}}$ and $P_{\text{min}}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$.
Answer: 585
user: On Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers). 

Define any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u < j_v$ for all $u,v \in \{1,2\}$ as a good group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.
Answer: 1009
user: Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$.
Answer: 121
user: In the figure on the right, $O$ is the center of the circle. The lines $OK$ and $OA$ are perpendicular to each other. $M$ is the midpoint of $OK$, and $BN$ is parallel to $OK$. Given that $\angle AMN = \angle NMO$, determine the measure of $\angle ABN$ in degrees.

```asy
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label("$A$", A, dir(O--A));
label("$K$", K, dir(O--K));
label("$B$", B, dir(O--B));
label("$N$", N, E);
label("$M$", M, S);
label("$O$", O, SE);
```

Note: The problem contains geometric constructs and angle relationships that need to be solved to find the measure of the specified angle. Ensure the Asymptote (asy) code is correctly rendered to visualize the problem's diagram.
Answer: 36
user: In a certain foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies within the government. Friendship and enmity are symmetric relations. A triplet of ministers is called **uniform** if all three are either mutual friends or mutual enemies. How many uniform triplets are there?
Answer: 40
user: Evan divides $100$ indivisible marbles among three volunteers: David, Justin, and Michael. Each volunteer must receive at least one marble. In the middle of the night, Lewis will select a positive integer $n > 1$ and attempt to steal exactly $\frac{1}{n}$ of each volunteer's marbles, provided $n$ divides the number of marbles they have. Determine the number of ways Evan can distribute the $100$ marbles such that Lewis cannot steal marbles from every volunteer, regardless of the $n$ he chooses.
Answer: 3540
user: The diagrams below show a $2 \times 2$ grid made up of four $1 \times 1$ squares. Two paths are illustrated on the grid, starting from the lower left corner to the upper right corner, with lengths $4$ and $6$ respectively. A path may not intersect itself by moving to a point where it has already been. Find the sum of the lengths of all possible paths from the lower left corner to the upper right corner of the grid.
Answer: 64
user: Let $A$ and $B$ be unit hexagons that share a center. Let $\mathcal{P}$ be the set of points contained in at least one of the hexagons. If the maximum possible area of $\mathcal{P}$ is $X$ and the minimum possible area of $\mathcal{P}$ is $Y$, then the value of $Y-X$ can be expressed as $\frac{a\sqrt{b}-c}{d}$, where $a, b, c, d$ are positive integers such that $b$ is square-free and $\gcd(a, c, d) = 1$. Find $a + b + c + d$.
Answer: 62
user: Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\} + [r] = 125$?
Answer: 2837
user: Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. For example, for his first card, Victor must draw the top card from one of the $3$ piles. In how many orders can Victor draw the cards?
Answer: 1680
user: There are circles $\omega_1$ and $\omega_2$. They intersect in two points, one of which is the point $A$. Point $B$ lies on $\omega_1$ such that $AB$ is tangent to $\omega_2$. The tangent to $\omega_1$ at $B$ intersects $\omega_2$ at points $C$ and $D$, where $D$ is closer to $B$. The line $AD$ intersects $\omega_1$ again at $E$. Given that $BD = 3$ and $CD = 13$, find the ratio $\frac{EB}{ED}$.
Answer: 4
user: The cost of five water bottles is $13, rounded to the nearest dollar, and the cost of six water bottles is $16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle.
Answer: 11
user: A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points?

\[\text{[asy]}\]
\[\text{draw}((0,-2)--\text{arc}((0,0),1,0,180)--\text{cycle});\]
\[\text{dot}((-0.8775,-0.245));\]
\[\text{dot}((-0.735,-0.53));\]
\[\text{dot}((-0.5305,-0.939));\]
\[\text{dot}((-0.3875,-1.225));\]
\[\text{dot}((-0.2365,-1.527));\]
\[\text{dot}((0.155,-1.69));\]
\[\text{dot}((0.306,-1.388));\]
\[\text{dot}((0.4,-1.2));\]
\[\text{dot}((0.551,-0.898));\]
\[\text{dot}((0.837,-0.326));\]
\[\text{dot}(\text{dir}(25));\]
\[\text{dot}(\text{dir}(50));\]
\[\text{dot}(\text{dir}(65));\]
\[\text{dot}(\text{dir}(100));\]
\[\text{dot}(\text{dir}(115));\]
\[\text{dot}(\text{dir}(140));\]
\[\text{[/asy]}\]
Answer: 540
user: A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.
Answer: 22
user: Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$.
Answer: 11
user: Let $\mathcal{A} = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a regular 2014-simplex, meaning the 2015 vertices of $\mathcal{A}$ lie in 2014-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when $PA_0^2 + PA_1^2 + \dots + PA_{2014}^2$ is divided by $10^6$.
Answer: 348595
user: Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 143, \\
(a+c)(b+d) &= 150, \\
(a+d)(b+c) &= 169.
\end{align*}
\]
Compute the smallest possible value of $a^2+b^2+c^2+d^2$.
Answer: 214
user: In quadrilateral $ABCD$, where $AC = BD$ and $\angle B = 60^\circ$, denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. Given that $MN = 12$ and the area of quadrilateral $ABCD$ is 420, compute $AC$.
Answer: 37
user: Young Guy likes to make friends with numbers, so he calls a number "friendly" if the sum of its digits is equal to the product of its digits. How many $3$-digit friendly numbers are there?
Answer: 6
user: Consider a $2 \times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?
Answer: 88
user: Let $f(x)$ be a monic cubic polynomial such that the sum of the coefficients of $f(x)$ is $5$ and the sum of the roots of $f(x)$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f(x)$.
Answer: 14
user: Find the sum of all positive divisors of $40081$.
Answer: 40500
user: Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.
Answer: 181440
user: **Problem 3.** A three-digit number ending in $2$ is written on the board. If we erase the $2$ and place it at the beginning of the number, the number decreases by $36$. What is the sum of the digits of the number?
Answer: 10
user: The diagram below shows an isosceles triangle with base $21$ and height $28$. Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square.
Answer: 150
user: Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.
Answer: 10
user: Let $\triangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D \neq A$ be the second intersection of $AB$ with $\omega$, and let point $E \neq A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$, where $a$, $b$ are relatively prime positive integers, find $a + b$.
Answer: 361
user: Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.
Answer: 69
user: On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallons?
Answer: 13
user: Given a positive integer $N$ (written in base $10$), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)

What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
Answer: 88888888
user: Find the number of ordered pairs of integers $(a,b) \in \{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a quadratic residue modulo $x^2+1$ and $35$. This means there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

- There exist polynomials $P(x)$, $Q(x)$ with integer coefficients such that $f(x)^2 - (ax+b) = (x^2+1)P(x) + 35Q(x)$.
- The remainder when the polynomial $f(x)^2 - (ax+b)$ is divided by the polynomial $x^2+1$ is a polynomial with integer coefficients, all divisible by $35$.
Answer: 225
user: Given that \( \frac{x}{\sqrt{x} + \sqrt{y}} = 18 \) and \( \frac{y}{\sqrt{x} + \sqrt{y}} = 2 \), find \( \sqrt{x} - \sqrt{y} \).
Answer: 16
user: What is the largest $n$ such that $n! + 1$ is a perfect square?
Answer: 7
user: A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime divisors of $n$.
Answer: 2
user: On a circular board, there are $19$ squares numbered in order from $1$ to $19$. To the right of $1$ is $2$, to the right of $2$ is $3$, and so on, until $1$ is to the right of $19$. In each box, there is a token. Every minute, each token moves to its right by the number of the box it is in at that moment plus one. For example, the token that is in the $7$th place moves the first minute $7 + 1$ places to its right to the $15$th square. The second minute, that same token moves $15 + 1$ places to the right, reaching square $12$, and so on. Determine if, at some point, all the tokens return to the place where they started, and if so, how many minutes must elapse.
Answer: 18
user: Find the minimum value of \( \sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)| \), where \( x, y \) are integers.
Answer: 2394000000
user: Find the least positive integer of the form $a\ b\ a\ a\ b\ a$, where $a$ and $b$ are distinct digits, such that the integer can be written as a product of six distinct primes.
Answer: 282282
user: Angeline starts with a 6-digit number and moves the last digit to the front. For example, if she originally had $100823$, she ends up with $310082$. Given that her new number is 4 times her original number, find the smallest possible value of her original number.
Answer: 102564
user: Find the smallest $n$ such that $2^{2000}$ divides $n!$.
Answer: 2008
user: Triangle $ABC$ satisfies $\angle ABC = \angle ACB = 78^\circ$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, and satisfy $\angle BCD = 24^\circ$ and $\angle CBE = 51^\circ$. If $\angle BED = x^\circ$, find $x$.
Answer: 12
user: Square $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1 \times 2$ rectangles are placed inside the square so that no two of the eight rectangles overlap. If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that none of the rectangles crosses the line segment $EF$. Find $m + n$.
Answer: 61
user: Determine the maximal possible length of the sequence of consecutive integers which can be expressed in the form $x^3 + 2y^2$, where $x$ and $y$ are integers.
Answer: 5
user: Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?
Answer: 10
user: There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position. When 1011 people walk by, they interact with the bulbs as follows:

1. The first person flips the switch on every bulb.
2. The second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.).
3. The third person flips the switch on every 5th bulb.
4. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$.

After all 1011 people have gone by, how many light bulbs are on?
Answer: 75
user: Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q$, and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in 10-dimensional space, it is possible to perform a finite number of jumps so that some two points coincide.
Answer: 1025