sungyub/dapo-math-17k-verl huggingface dataset:
user: Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:

\[
1 < a < b+2 < 10.
\]

Answer: 28
user: Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.

Answer: 865
user: Triangle $ABC$ has $AB = BC = 10$ and $CA = 16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$.

Answer: 24
user: Consider trapezoid $ABCD$ with $AB \parallel CD$, where $AB = 5$ and $CD = 9$. Additionally, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$?

Answer: 2
user: The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$.

Answer: 9
user: Maryssa, Stephen, and Cynthia played a game. Each of them independently privately chose one of Rock, Paper, and Scissors at random, with all three choices being equally likely. Given that at least one of them chose Rock and at most one of them chose Paper, the probability that exactly one of them chose Scissors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 916
user: Let $p$, $q$, and $r$ be the three roots of the polynomial $x^3 - 2x^2 + 3x - 2023$. Suppose that the polynomial $x^3 + Bx^2 + Mx + T$ has roots $p + q$, $p + r$, and $q + r$ for real numbers $B$, $M$, and $T$. Compute $B - M + T$.

Answer: 2006
user: In triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.

Answer: 77
user: Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 107
user: Five people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *trio* if one of the following conditions is true:

$A$ shakes hands with $B$, and $B$ shakes hands with $C$, or
$A$ does not shake hands with $B$, and $B$ does not shake hands with $C$.

If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.

Answer: 10
user: Let $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, ... be the sequence of prime numbers. Find the least positive even integer $n$ such that $p_1 + p_2 + p_3 + \ldots + p_n$ is not a prime number.

Answer: 8
user: Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.

Answer: 248
user: In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, the relationship is defined as:
$$ S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}. $$
For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on.

Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?

Answer: -2
user: Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, individually and simultaneously.

Find the largest positive integer $k$ such that Dexter can set up the programs so that, no matter how the numbers are distributed, there are always at least $k$ robots who guess their numbers correctly.

Answer: 17
user: Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.

Answer: 4027
user: The sequence $x_1, x_2, x_3, \ldots$ is defined by $x_1 = 2022$ and $x_{n+1} = 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of $7$ for any positive integer $n$.

Answer: 404
user: How many lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in its interior?

Answer: 1343
user: The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved?

$$\frac{\frac{\square}{\square} + \frac{\square}{\square}}{\frac{\square}{\square}}$$

Answer: 23
user: Consider the sequence $2, 3, 5, 6, 7, 8, 10, \ldots$ of all positive integers that are not perfect squares. Determine the $2011^{\text{th}}$ term of the sequence.

Answer: 2056
user: If $S$ is a set consisting of $12$ elements, what is the maximum number of pairs $(a, b)$ such that $a, b \in S$ and $\frac{b}{a}$ is a prime number?

Answer: 20
user: Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$.

Answer: 28
user: Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A pseudo-ultrafilter $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm}(a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?

Answer: 19
user: Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Compute $m+n$.

Answer: 751
user: For all positive integers $k$, define $f(k) = k^2 + k + 1$. Compute the largest positive integer $n$ such that:
\[
2015 \cdot f(1^2) \cdot f(2^2) \cdots f(n^2) \geq \left(f(1) \cdot f(2) \cdots f(n)\right)^2.
\]

Answer: 44
user: Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.

Answer: 2002
user: Find the greatest possible value of $pq + r$, where $p$, $q$, and $r$ are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

Answer: 1008
user: Find the number of nondegenerate triangles whose vertices lie in the set of points $(s, t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, where $s$ and $t$ are integers.

Answer: 2148
user: Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\frac{1}{3}$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.

Answer: 245
user: Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Answer: 43
user: Let \( C = \{ (i,j) | i,j \text{ integers such that } 0 \leq i,j \leq 24 \} \). How many squares can be formed in the plane such that all of their vertices are in \( C \) and their sides are parallel to the \( X \)-axis and \( Y \)-axis?

Answer: 4900
user: Given a regular hexagon $ABCDEF$, let point $P$ be the intersection of lines $BC$ and $DE$, and let point $Q$ be the intersection of lines $AP$ and $CD$. If the area of $\triangle QEP$ is equal to $72$, find the area of regular hexagon $ABCDEF$.

Answer: 324
user: The Matini company released a special album with the flags of the $12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope contains two flags chosen randomly. Determine the minimum number of envelopes that need to be opened so that the probability of having a repeated flag is $50\%$.

Answer: 3
user: Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.

Answer: 5
user: Say a positive integer $n$ is radioactive if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$, and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$?

Answer: 119
user: For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) = n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.

Answer: 15
user: Determine the smallest natural number written in the decimal system such that the product of its digits equals $10! = 1 \times 2 \times 3 \times \ldots \times 9 \times 10$.

Answer: 45578899
user: The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$.

Answer: 18
user: Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

Answer: 20
user: Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.

Answer: 8
user: On equilateral $\triangle{ABC}$, point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J, which are the vertices of another equilateral triangle. The area of $\triangle{ABC}$ is twice the area of $\triangle{GHJ}$. The side length of $\triangle{ABC}$ can be written as $\frac{r+\sqrt{s}}{t}$, where r, s, and t are relatively prime positive integers. Find $r + s + t$.

Answer: 30
user: Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.

Answer: 2
user: The sequence of integers $a_1, a_2, \dots$ is defined as follows:
- $a_1 = 1$.
- For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different.

Define $a_{2004}$.

Answer: 4507
user: For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) + \cos^2(30^\circ +n)$. Evaluate $4\sum^{30}_{n=1} n \cdot T(n).$

Answer: 1395
user: Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$?

Answer: 8
user: Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}
Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Compute $100m + n$.

Answer: 2105
user: Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

Answer: 18
user: Find the least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$.

Answer: 10
user: Determine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two given or already obtained powers of the number $x$. For example, $x \cdot x = x^{2}$, $x^{2} \cdot x^{2} = x^{4}$, $x^{4} : x = x^{3}$, etc.

Answer: 14
user: Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S$.

Answer: 64
user: Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y)$, such that $f(0,k) = f(k,0) = 2^k$ and 
\[ f(a,b) + f(a+1,b+1) = f(a+1,b) + f(a,b+1) \]
for all nonnegative integers $a, b$. Determine the number of positive integers $n \leq 2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b) = n$.

Answer: 65
user: Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n > 18$:
\[
\begin{align*}
x + 3y & \equiv 7 \pmod{n} \\
2x + 2y & \equiv 18 \pmod{n} \\
3x + y & \equiv 7 \pmod{n}
\end{align*}
\]
Compute the smallest nonnegative integer $a$ such that $2x \equiv a \pmod{n}$.

Answer: 20
user: Let $\triangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\triangle ABC$ again at the point $M \neq A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$. Compute $AX \cdot AY$.

Answer: 36
user: Let $a$ and $b$ be positive integers such that $1 \leq a < b \leq 100$. If there exists a positive integer $k$ such that $ab \mid a^k + b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.

Answer: 96
user: Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$. 

A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.

Answer: 961
user: Find the least natural number $n$ such that, if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

Answer: 96
user: Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?

Answer: 90
user: In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. If $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, where $m, n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

Answer: 34
user: How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy 
\[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \]

Answer: 252
user: Find the number of 5-digit numbers that each contains the block '15' and is divisible by 15.

Answer: 479
user: Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Answer: 1010
user: Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, find $A$.

Answer: 344
user: Five students take a test where any integer score from $0$ to $100$ is possible. What is the largest possible difference between the median and the mean of the scores?

Answer: 40
user: Let $C$ denote the curve $y^2 = \frac{x(x+1)(2x+1)}{6}$. The points $(\frac{1}{2}, a)$, $(b, c)$, and $(24, d)$ lie on $C$ and are collinear, and $ad < 0$. Given that $b$ and $c$ are rational numbers, find $100b^2 + c^2$.

Answer: 101
user: Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each square, a number will be written based on the following rules:

- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Answer: 486
user: Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$?

Answer: 333
user: Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$, and the graphs of the functions $y = \sqrt{x^3 + 1}$ and $y = -\sqrt[3]{x^2 + 2x}$.

Answer: 6
user: How many ordered pairs of integers $(x, y)$ satisfy the equation: 

$$2x^2 + y^2 + xy = 2(x + y)$$

Answer: 4
user: Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.

Answer: 335
user: Find the sum of all three-digit numbers (written in base $10$) such that the leading digit is the sum of the other two digits. Express your answer in base $10$.

Answer: 34815
user: Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.

Answer: 7
user: Let $a$ and $b$ be nonzero real numbers such that $\frac{1}{3a} + \frac{1}{b} = 2011$ and $\frac{1}{a} + \frac{1}{3b} = 1$. What is the quotient when $a + b$ is divided by $ab$?

Answer: 1509
user: Consider the function $f(x) = 5x^4 - 12x^3 + 30x^2 - 12x + 5$. Let $f(x_1) = p$, where $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.

Answer: 5
user: Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$.

Answer: 9999
user: Ryan chooses five subsets $S_1, S_2, S_3, S_4, S_5$ of the set $\{1, 2, 3, 4, 5, 6, 7\}$ such that $|S_1| = 1$, $|S_2| = 2$, $|S_3| = 3$, $|S_4| = 4$, and $|S_5| = 5$. Moreover, for all $1 \le i < j \le 5$, either $S_i \cap S_j = S_i$ or $S_i \cap S_j = \emptyset$ (in other words, the intersection of $S_i$ and $S_j$ is either $S_i$ or the empty set). In how many ways can Ryan select the sets?

Answer: 11760
user: There is a positive integer $s$ such that there are $s$ solutions to the equation
\[ 64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46 \]
in the interval \( (0, \frac{\pi}{2}) \), all of the form \( \frac{m_k}{n_k}\pi \), where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, \ldots, s$. Find \((m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + \cdots + (m_s + n_s)\).

Answer: 100
user: Let $f(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x)$ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2 \rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.

Answer: 388
user: Find the greatest positive integer $n$ such that $n$ is divisible by all positive integers whose cube is not greater than $n$.

Answer: 420
user: Consider all pairs of points $(a,b,c)$ and $(d,e,f)$ in the 3-D coordinate system with the condition $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?

Answer: 90
user: Find the smallest integer $n \geq 1$ such that the equation:
$$a^2 + b^2 + c^2 - nd^2 = 0$$
has $(0, 0, 0, 0)$ as the unique solution.

Answer: 7
user: Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.

Answer: 42
user: Let $P(x)$ be a real quadratic trinomial such that for all $x \in \mathbb{R}$, the inequality $P(x^3 + x) \geq P(x^2 + 1)$ holds. Find the sum of the roots of $P(x)$.

Answer: 4
user: As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. 

```asy
size(120); defaultpen(linewidth(0.7)+fontsize(10)); 
pair D2(real x, real y) {
 pair P = (x,y);
 dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label("$A$",D2(0,big),NW);
label("$B$",D2(0,0),SW);
label("$C$",D2(big,0),SW);
label("$D$",D2(big,big),N);
label("$E$",D2(big+small,0),SE);
label("$F$",D2(big+small,small),NE);
label("$G$",D2(big,small),NE);
```

Unnecessary information like author or edit history has been removed.

Answer: 98
user: Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart.

Answer: 10
user: The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.

Answer: 638
user: How many ordered pairs of integers \((a,b)\) satisfy all of the following inequalities? 

\[ \begin{align*} 
& a^2 + b^2 < 16 \\
& a^2 + b^2 < 8a \\
& a^2 + b^2 < 8b 
\end{align*} \]

Answer: 6
user: In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.

Answer: 91
user: Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?

Answer: 25