sungyub/dapo-math-17k-verl huggingface dataset:
user: There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Answer: 1009
user: 162 pluses and 144 minuses are placed in a $30 \times 30$ table such that each row and each column contains at most 17 signs. (No cell contains more than one sign.) For every plus, count the number of minuses in its row, and for every minus, count the number of pluses in its column. Find the maximum of the sum of these numbers.

Answer: 2592
user: Find the sum of all positive integers $1 \leq k \leq 99$ such that there exist positive integers $a$ and $b$ with the property that 
\[ x^{100} - ax^k + b = (x^2 - 2x + 1)P(x) \] 
for some polynomial $P(x)$ with integer coefficients.

Answer: 117
user: Let $a, b, c$, and $d$ be real numbers such that:
\[ a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018. \]
Evaluate the expression:
\[ 3b + 8c + 24d + 37a. \]

Answer: 1215
user: Find the smallest positive integer $n$ which cannot be expressed as \( n = \frac{2^a - 2^b}{2^c - 2^d} \) for some positive integers \( a, b, c, \) and \( d \).

Answer: 11
user: Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?

Answer: 20
user: Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?

Answer: 509
user: Find the smallest positive real $t$ such that the system of equations
\[\begin{align*}
 x_1 + x_3 &= 2t x_2, \\
 x_2 + x_4 &= 2t x_3, \\
 x_3 + x_5 &= 2t x_4 \\
\end{align*}\]
has a solution $x_1, x_2, x_3, x_4, x_5$ in non-negative reals, not all zero.

Answer: 1
user: In a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?

Answer: 1010
user: Find the number of sets $A$ that satisfy the following conditions:

$A$ is a set of two positive integers.
Each number in $A$ is at least $22\%$ the size of the other number.
$A$ contains the number $30$.

Answer: 129
user: Compute the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.

Answer: 15
user: Quadrilateral $ABCD$ has integer side lengths, and angles $\angle ABC$, $\angle ACD$, and $\angle BAD$ are right angles. Compute the smallest possible value of $AD$.

Answer: 25
user: Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or the number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "very prime" number with justification.

Answer: 373
user: Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.

Answer: 22
user: Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.

Answer: 274
user: Compute the number of permutations of \(\{1,2,3\}\) with the property that there is some number that can be removed such that the remaining numbers are in increasing order. For example, \((2,1,3)\) has this property because removing \(1\) leaves \((2,3)\), which is in increasing order.

Answer: 5
user: Let $ABC$ be a triangle with $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m + \sqrt{n}$ for positive integers $m$ and $n$. What is $100m + n$?

Answer: 460
user: Let $a_1,a_2,\ldots, a_{2013}$ be a permutation of the numbers from $1$ to $2013$. Let $A_n = \frac{a_1 + a_2 + \cdots + a_n}{n}$ for $n = 1,2,\ldots, 2013$. If the smallest possible difference between the largest and smallest values of $A_1,A_2,\ldots, A_{2013}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Answer: 3
user: Find the greatest natural number $n$ such that there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural numbers $a_{1} < a_{2} < \ldots < a_{n-1}$ satisfying the following equations for $i = 1, 2, \ldots, n-1$: 
\[x_{1}x_{2}\ldots x_{n} = 1980 \quad \text{and} \quad x_{i} + \frac{1980}{x_{i}} = a_{i}.\]

Answer: 6
user: Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.

Answer: 186
user: Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.

Answer: 729
user: What are the last two digits of the number $(11^2 + 15^2 + 19^2 + \ldots + 2007^2)^2$?

Answer: 0
user: Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\frac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.

Answer: 26
user: Let $f(x)$ be the function such that $f(x) > 0$ for $x \geq 0$ and $\{f(x)\}^{2006} = \int_{0}^{x} f(t) \, dt + 1.$

Find the value of $\{f(2006)\}^{2005}.$

Answer: 2006
user: Compute the number of ordered triples of integers $(a, b, c)$ between $1$ and $12$, inclusive, such that, if \[ q = a + \frac{1}{b} - \frac{1}{b + \frac{1}{c}}, \] then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.

Answer: 132
user: A circle is divided into $n$ equal parts. Marceline sets out to assign whole numbers from $1$ to $n$ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $887$, $217$, and $1556$ occupy consecutive positions. How many parts was the circumference divided into?

Answer: 2009
user: Determine the minimum value of the sum of several distinct positive integers whose product is divisible by $2006^2$.

Answer: 228
user: Let $(p_1, p_2, \dots) = (2, 3, \dots)$ be the list of all prime numbers, and $(c_1, c_2, \dots) = (4, 6, \dots)$ be the list of all composite numbers, both in increasing order. Compute the sum of all positive integers $n$ such that $|p_n - c_n| < 3$.

Answer: 16
user: Compute $\log_2 6 \cdot \log_3 72 - \log_2 9 - \log_3 8$.

Answer: 5
user: Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying the relation $4f(3-x) + 3f(x) = x^2$ for any real $x$. Find the value of $f(27) - f(25)$ to the nearest integer. (Here $\mathbb{R}$ denotes the set of real numbers.)

Answer: 8
user: Find the sum of all real solutions to $(x^2 - 10x - 12)^{x^2+5x+2} = 1$.

Answer: 15
user: The vertices of hexagon $ABCDEF$ lie on a circle. The sides of the hexagon are given as follows: $AB = CD = EF = 6$ and $BC = DE = FA = 10$. The area of the hexagon is $m\sqrt{3}$. Find the value of $m$.

Answer: 94
user: Wu starts out with exactly one coin. Wu flips every coin he has at once after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$.

Answer: 71622400
user: $x$, $y$, and $z$ are positive real numbers that satisfy $x^3 + 2y^3 + 6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.

Answer: 35
user: Let $S$ be the set of all pairs of positive integers $(x, y)$ for which $2x^2 + 5y^2 \le 5 + 6xy$. Compute $\displaystyle\sum_{(x,y) \in S} (x+y+100)$.

Answer: 839
user: Consider a $2 \times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
\[
\begin{bmatrix} 
1 & 2 & 0 \\
2 & 1 & 0 
\end{bmatrix}
\]

Answer: 9
user: If $2011^{2011^{2012}} = x^x$ for some positive integer $x$, how many positive integer factors does $x$ have?

Answer: 2012
user: Find the number of 6-tuples $(x_1, x_2, \ldots, x_6)$, where $x_i = 0, 1, \text{ or } 2$ and $x_1 + x_2 + \cdots + x_6$ is even.

Answer: 365
user: The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$ 

Alice chooses two numbers $a,b,$ erases them, and writes the number $\frac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.

If the sum of all possible numbers she can end up with is $\frac{p}{q}$ where $p, q$ are coprime, then what is the value of $\log_{2}(p+q)$?

Answer: 100
user: By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?

Answer: 21600
user: Let $V$ be a rectangular prism with integer side lengths. The largest face has an area of $240$ and the smallest face has an area of $48$. A third face has an area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

Answer: 260
user: Aida made three cubes with positive integer side lengths $a$, $b$, and $c$. She decided to dismantle these cubes into unit cubes and tried to construct a larger cube with side length $a+b+c$. Unfortunately, she was 648 blocks short. How many possible ordered triples $(a, b, c)$ exist?

Answer: 7
user: I have 6 friends and during a vacation, I met them during several dinners. I found that:

- I dined with all 6 friends exactly on 1 day.
- I dined with every 5 of them on 2 days.
- I dined with every 4 of them on 3 days.
- I dined with every 3 of them on 4 days.
- I dined with every 2 of them on 5 days.

Additionally, every friend was present at 7 dinners and absent at 7 dinners. How many dinners did I have alone?

Answer: 1
user: The product $20! \cdot 21! \cdot 22! \cdot \ldots \cdot 28!$ can be expressed in the form $m \cdot n^3$, where $m$ and $n$ are positive integers, and $m$ is not divisible by the cube of any prime. Find $m$.

Answer: 825
user: How many ways are there to color the vertices of a triangle using the colors red, green, blue, or yellow such that no two vertices have the same color? Note that rotations and reflections are considered distinct.

Answer: 24
user: Compute the area of the trapezoid $ABCD$ with right angles $\angle BAD$ and $\angle ADC$. The side lengths are given as $AB = 3$, $BC = 5$, and $CD = 7$.

Answer: 15
user: Find the $2019^{th}$ strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

Answer: 37805
user: For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\frac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.

Answer: 11
user: Let $a_0 = 1$, $a_1 = 2$, and $a_n = 4a_{n-1} - a_{n-2}$ for $n \ge 2$. Find an odd prime factor of $a_{2015}$.

Answer: 181
user: Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round, a winner is determined on each court. After each round:

The player who lost on court $1$ and the player who won on court $20$ remain in their respective positions.
For the remaining $38$ players:
 - The winner on court $i$ moves to court $i + 1$.
 - The loser moves to court $i - 1$.

The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

Answer: 39
user: Point $Y$ lies on line segment $XZ$ such that $XY = 5$ and $YZ = 3$. Point $G$ lies on line $XZ$ such that there exists a triangle $ABC$ with centroid $G$ where $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$. Compute the largest possible value of $XG$.

Answer: 2
user: Let $S = \{1, 2, 3, \ldots, 12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?

Answer: 31
user: Find the value of $a$ such that:

\[101a = 6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]

Answer: 2012
user: Among the coordinates $(x,y)$ where $1 \leq x,y \leq 101$, choose some points such that there do not exist $4$ points which form an isosceles trapezium with its base parallel to either the $x$ or $y$ axis (including rectangles). Find the maximum number of coordinate points that can be chosen.

Answer: 201
user: Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order.

The following action is performed repeatedly (any number of times):

\[\text{def action():} \]
\[\text{Choose an integer } n \text{ between } 1 \text{ and } 10 \text{ inclusive} \]
\[\text{Reverse the array between indices } 1 \text{ and } n \text{ inclusive} \]
\[\text{Reverse the array between indices } n+1 \text{ and } 10 \text{ inclusive (If } n = 10, \text{ we do nothing)} \]

How many possible orders can the array have after we are done with this process?

Answer: 20
user: Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

Answer: 217
user: Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of $105$ free throws between them, with each person taking at least one free throw. If Andrew made exactly $\frac{1}{3}$ of his free throw attempts and Beatrice made exactly $\frac{3}{5}$ of her free throw attempts, what is the highest number of successful free throws they could have made between them?

Answer: 59
user: Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? 

(The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)

Answer: 40
user: $F(0) = 3$ and $F(n) = F(n-1) + 4$ when $n$ is positive. Find $F(F(F(5)))$.

Answer: 383
user: Let \( \{x\} \) denote the fractional part of \( x \). For example, \( \{5.5\} = 0.5 \). Find the smallest prime \( p \) such that the inequality \[ \sum_{n=1}^{p^2}\left\{\frac{n^p}{p^2}\right\} > 2016 \] holds.

Answer: 67
user: Define the minimum real $C$ such that for any real numbers $0 = a_0 < a_1 < \dots < a_{1000}$, the inequality $$\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor$.

Answer: 127
user: There are $2022$ equally spaced points on a circular track $\gamma$ with a circumference of $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labelings of the $2022$ points.

Answer: 2042222
user: A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

Answer: 4
user: The diagram below shows an equilateral triangle and a square with a side length of $2$ joined along an edge. What is the area of the shaded triangle?

```latex
\begin{asy}
fill((2,0) -- (2,2) -- (1, 2 + \sqrt{3}) -- cycle, gray);
draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + \sqrt{3}) -- (0,2) -- (0,0));
draw((0,2) -- (2,2));
\end{asy}
```

Answer: 1
user: How many six-letter words formed from the letters of "AMC" do not contain the substring "AMC"? For example, "AMAMMC" has this property, but "AAMCCC" does not.

Answer: 622
user: The sides and vertices of a pentagon are labeled with the numbers $1$ through $10$. Each side of the pentagon must have the same sum of numbers. What is the smallest possible value of this sum?

Answer: 14
user: The fraction $\frac{1}{2015}$ has a unique "(restricted) partial fraction decomposition" of the form \[\frac{1}{2015} = \frac{a}{5} + \frac{b}{13} + \frac{c}{31},\] where $a$, $b$, and $c$ are integers with $0 \leq a < 5$ and $0 \leq b < 13$. Find $a + b$.

Answer: 14
user: For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation 
\[4096f(f(x, x), x) = x^{13}\] 
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.

Answer: 177
user: Two circles have radii of 5 and 26, respectively. The smaller circle passes through the center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?

Answer: 4
user: Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$.

Answer: 7294927
user: On a large wooden block, there are four twelve-hour analog clocks of varying accuracy. At 7 PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7 PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.)

Answer: 12
user: For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. Once a sapling is turned into a perfectly-shaped apple tree, it will remain so. 

The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Answer: 789953
user: Let $a$, $b$, $c$, $d$, $e$, $f$, and $g$ be seven distinct positive integers not bigger than 7. Find all primes that can be expressed as $abcd + efg$.

Answer: 179
user: Find the sum of the four least positive integers each of whose digits add to $12$.

Answer: 210
user: A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2 - 4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.

Answer: 6900
user: Find all integer solutions of the equation:

\[
\left\lfloor \frac{x}{1!} \right\rfloor + \left\lfloor \frac{x}{2!} \right\rfloor + \cdots + \left\lfloor \frac{x}{10!} \right\rfloor = 1001.
\]

Answer: 584
user: Let $ABCD$ be a rhombus with sides $AB = BC = CD = DA = 13$. Construct another rhombus $BAFE$ on the side $AB$ outside $ABCD$, such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

Answer: 120
user: Let $x$ and $y$ be positive real numbers. Define $a = 1 + \frac{x}{y}$ and $b = 1 + \frac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.

Answer: 50
user: A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 11
user: The positive integer $m$ is a multiple of $111$, and the positive integer $n$ is a multiple of $31$. Their sum is $2017$. Find $n - m$.

Answer: 463
user: Consider a cyclic quadrilateral with side lengths $1$, $4$, $8$, and $7$ in that order. What is its circumdiameter? Let the answer be in the form $a\sqrt{b} + c$, where $b$ is squarefree. Find the value of $a + b + c$.

Answer: 66
user: Let $f(x) = x^3 + ax + b$, where $a \neq b$. Suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$.

Answer: 1
user: On a natural number $n$, you are allowed two operations: 
Multiply $n$ by $2$, or 
Subtract $3$ from $n$. 

For example, starting with $8$, you can reach $13$ as follows: $8 \longrightarrow 16 \longrightarrow 13$. You need two steps, and you cannot do it in less than two steps. 

Starting from $11$, what is the least number of steps required to reach $121$?

Answer: 10
user: Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \frac{1}{2}| + |y - \frac{1}{2}| < \frac{1}{2}$ is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$

Answer: 3
user: Let $f(x) = x^2 + x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f\left(\frac{1}{y}\right) + f\left(\frac{1}{z}\right) = \frac{1}{10}$. Compute $100m + n$.

Answer: 1735
user: Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\triangle MNP$ can be written in simplest form as $\frac{a}{b}$. Find $a + b$.

Answer: 457
user: Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e., it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?

Answer: 70
user: Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.

Answer: 1481480
user: Find the remainder when $2020!$ is divided by $2020^2$.

Answer: 0