sungyub/dapo-math-17k-verl huggingface dataset:
user: Find the integer closest to the value of the following expression:

$$\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right)^{2023} \right)$$

Answer: 0
user: Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$

Answer: 12
user: The integers from $1$ to $25$, inclusive, are randomly placed into a $5 \times 5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1, 2, 3, 4,$ and $5$, then the expected column number of the entry $23$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.

Answer: 17
user: Consider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.

Answer: 169
user: Let $f$ be a function defined on the set of positive integers, with values in the same set, which satisfies:

- $f(n + f(n)) = 1$ for all $n \ge 1$.
- $f(1998) = 2$

Find the lowest possible value of the sum $f(1) + f(2) + \ldots + f(1999)$, and find the formula of $f$ for which this minimum is satisfied.

Answer: 1997003
user: Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ again at $A_1$ and $B_1$, respectively, such that $A_1$ and $B_1$ are on opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$, and $OP = 10$, find the radius of $\Omega_1$.

Answer: 20
user: $x$, $y$, and $z$ are real numbers such that $xyz = 10$. What is the maximum possible value of $x^3 y^3 z^3 - 3x^4 - 12y^2 - 12z^4$?

Answer: 760
user: The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 47
user: Determine the last two digits of the number $Q = 2^{2017} + 2017^2$.

Answer: 61
user: How many real numbers $x$ are solutions to the equation $|x - 2| - 4 = \frac{1}{|x - 3|}$?

Answer: 2
user: There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds. 

- Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction with the same speed.
- Whenever they meet face-to-face, the faster one instantly turns around and starts running in the opposite direction with the same speed.

Determine how far from the marked point their 100th meeting could occur.

Answer: 0
user: There are $2010$ people sitting around a round table. First, we give one person $x$ a candy. Next, we give candies to the $1^{st}$ person, $1+2^{nd}$ person, $1+2+3^{rd}$ person, $\cdots$, and $1+2+\cdots+2009^{th}$ person clockwise from $x$. Find the number of people who get at least one candy.

Answer: 408
user: From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number?

Answer: 358926471
user: Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box?

```
\begin{asy}
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));

draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
\end{asy}
```

(Note: Removed extraneous information like author details and unnecessary formatting instructions.)

Answer: 118
user: The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ and $y$, erases them, and then writes down $x + y + xy$. He continues this process until only one number remains on the blackboard. What is this number?

Answer: 2010
user: Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A \cdot B$?

Answer: 1811716
user: A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems.

We create an \( 8 \times 4 \) table. Inside the square located in the \( i \)-th row and \( j \)-th column, we write down the number of people who correctly solved both problem \( S_i \) and \( F_j \). If the 32 entries in the table sum to 256, what is the value of \( n \)?

Answer: 10
user: Find the greatest value of $C$ for which, for any $x, y, z, u$ such that $0 \le x \le y \le z \le u$, the inequality holds:
$$(x + y + z + u)^2 \ge C yz.$$

Answer: 9
user: Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.

Answer: 750
user: Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Answer: 390
user: Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$?

Answer: 10071
user: Consider the addition problem:
\[
\begin{array}{ccccc}
 & C & A & S & H \\
+ & & & M & E \\
\hline
 O & S & I & D & E
\end{array}
\]
where each letter represents a base-ten digit, and \(C, M, O \neq 0\). Distinct letters are allowed to represent the same digit. How many ways are there to assign values to the letters so that the addition problem is true?

Answer: 0
user: In the obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $CA = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD] + [BCD]$. 

(Note that $[XYZ]$ denotes the area of triangle $XYZ$.)

Answer: 150
user: Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

Answer: 35
user: There exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.

Answer: 2001
user: The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

Answer: 1028
user: Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even.

Answer: 150
user: Find the smallest natural number $x > 0$ such that all the following fractions are simplified:

$$\frac{3x+9}{8}, \frac{3x+10}{9}, \frac{3x+11}{10}, \ldots, \frac{3x+49}{48}$$

This means that the numerators and denominators are relatively prime.

Answer: 20
user: Determine all positive integers that are equal to $300$ times the sum of their digits.

Answer: 2700
user: Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.

Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and 
\[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.

Answer: 10010
user: Find the minimum positive value of $1 \times 2 \times 3 \times \ldots \times 2022$ where you can replace $\times$ with $+$ or $-$.

Answer: 1
user: Find the number of pairs \((a, b)\) of positive integers such that the greatest common divisor (GCD) of \(a\) and \(b\) is equal to \(1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50\), and the least common multiple (LCM) of \(a\) and \(b\) is \(1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^2\).

Answer: 32768
user: Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.

Answer: 779
user: Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. For example, $2^1$ divides $2150$. Note: The number may start with $0$.

Answer: 10
user: Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.

Answer: 16
user: Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got $+50$ (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $[90, \infty)$ gets an A, a score in the range $[80, 90)$ gets a B, and a score in the range $[70, 80)$ gets a C. After the curve, Mr. Vader makes this statement: "Guess A, get an A. Guess B, get a B. Guess C, get a C." That is, answering every question with the answer choice $X$ would give, with the curve, a score receiving a grade of $X$, where $X$ is one of A, B, C. Luke, a student in Mr. Vader's class, was told ahead of time that there were either $5$ or $6$ answers as A on the test. Find the sum of all possible values of the number of questions on the test, given this information.

Answer: 64
user: A sequence $a_0, a_1, a_2, \dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each positive integer $n$. Compute $\left\lvert a_1 + a_2 + \dots + a_{1000} \right\rvert$.

Answer: 1332
user: The \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil$, then the first $\lceil \frac{2n}{3} \rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.

Answer: 243
user: The diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has an area of $100$, find the area of the entire triangle.

Answer: 640
user: Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$.

Answer: 1006
user: For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that Bekhzod summed $2024$ non-negative consecutive integers, Ozod summed $2024+x$ non-negative consecutive integers and they got the same result?

Answer: 1138
user: Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

Answer: 4
user: Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?

Answer: 15
user: Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts with $2012$. For example, $586$ is a leader since $586^3 = 201230056$. How many leaders are there in the set $\{1, 2, 3, \ldots, 2012\}$?

Answer: 2008
user: A subset $S$ of the set of integers $0$ to $99$ is said to have property $A$ if it is impossible to fill a crossword-puzzle with $2$ rows and $2$ columns using numbers in $S$ (where $0$ is written as $00$, $1$ as $01$, and so on). Determine the maximal number of elements in the set $S$ with the property $A$.

Answer: 25
user: Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.

Answer: 365
user: A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?

Answer: 18
user: What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?

Answer: 36
user: Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.

Answer: 99
user: When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?

\[ \textbf{(A) } 146 \quad \textbf{(B) } 147 \quad \textbf{(C) } 148 \quad \textbf{(D) } 149 \quad \textbf{(E) } 150 \]

Answer: 149
user: Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the \textit{floor} function, which returns the greatest integer less than or equal to $x$.

Answer: 1973
user: There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars?

Answer: 1012
user: It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4?

Answer: 32
user: Find the period of the repetend of the fraction $\frac{39}{1428}$ by using binary numbers, i.e., its binary decimal representation.

(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the repetend of the fraction, and the smallest length of the repetend, $k$, is called the period of the decimal number.)

Answer: 24
user: Let $x$ be a real number such that:
\[ 4^{2x} + 2^{-x} + 1 = (129 + 8\sqrt{2})(4^{x} + 2^{-x} - 2^{x}). \]
Find $10x$.

Answer: 35
user: Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k} a^2_i.\)

Answer: 476
user: In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter.

Answer: 77
user: The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions.

Answer: 200
user: It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.

Answer: 14
user: A week ago, Sandy's seasonal Little League batting average was $360$. After five more at bats this week, Sandy's batting average increased to $400$. What is the smallest number of hits that Sandy could have had this season?

Answer: 12
user: Let $x$ be a positive real number. Define
\[
 A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
 B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad
 C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] 
Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.

Answer: 183
user: How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.

Answer: 1
user: Let \( G, O, D, I, \) and \( T \) be digits that satisfy the following equation:

\[
\begin{array}{cccc}
 & G & O & G & O \\
+ & D & I & D & I \\
\hline
G & O & D & O & T
\end{array}
\]

(Note that \( G \) and \( D \) cannot be \( 0 \), and that the five variables are not necessarily different.)

Compute the value of GODOT.

Answer: 10908
user: A classroom has $30$ students, each of whom is either male or female. For every student $S$, we define their ratio to be the number of students of the opposite gender as $S$ divided by the number of students of the same gender as $S$ (including $S$). Let $\Sigma$ denote the sum of the ratios of all $30$ students. Find the number of possible values of $\Sigma$.

Answer: 2
user: There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

Answer: 50
user: In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $36\pi$, find the perimeter of the triangle.

Answer: 36
user: Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_{15})-15$, $P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.

Answer: 460
user: Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.

Answer: 65819
user: A function $f$ has its domain equal to the set of integers $0, 1, \ldots, 11$, and $f(n) \geq 0$ for all such $n$. The function $f$ satisfies the following conditions:
- $f(0) = 0$
- $f(6) = 1$
- If $x \geq 0$, $y \geq 0$, and $x + y \leq 11$, then $f(x + y) = \frac{f(x) + f(y)}{1 - f(x)f(y)}$

Find $f(2)^2 + f(10)^2$.

Answer: 14
user: Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

Answer: 9
user: Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.

Answer: 8
user: Suppose $2015 = a_1 < a_2 < a_3 < \cdots < a_k$ is a finite sequence of positive integers, and for all $m, n \in \mathbb{N}$ and $1 \le m, n \le k$, the inequality $$a_m + a_n \ge a_{m+n} + |m-n|$$ holds. Determine the largest possible value $k$ can obtain.

Answer: 2016
user: Let $\{a_n\}$ be a sequence of integers satisfying the recurrence relation:
\[
(n-1)a_{n+1} = (n+1)a_n - 2(n-1) \quad \forall n \ge 1.
\]
Given that $2000 \mid a_{1999}$, determine the smallest integer $n \ge 2$ such that $2000 \mid a_n$.

Answer: 249
user: Find the smallest constant $c$ such that for every 4 points in a unit square, there are two points with a distance less than or equal to $c$ apart.

Answer: 1
user: For how many integers $1 \leq k \leq 2013$ does the decimal representation of $k^k$ end with a $1$?

Answer: 202
user: Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take?

Answer: 81
user: An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both. If there is no such square, consider it as if it were orange. The only exception is the bottom left square, which is black. Consider the diagonal from the upper left to the lower right. How many black squares does this diagonal have?

Answer: 40
user: The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.

Answer: 74