sungyub/dapo-math-17k-verl huggingface dataset:
user: The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.

```latex
\begin{asy}
 unitsize(5);
 pair A = (0, 20 * 21) / 29.0;
 pair B = (-20^2, 0) / 29.0;
 pair C = (21^2, 0) / 29.0;
 draw(A -- B -- C -- cycle);
 label("$A$", A, S);
 label("$B$", B, S);
 label("$C$", C, S);
 filldraw(arc((A + C)/2, C, A)--cycle, gray);
 filldraw(arc((B + C)/2, C, A)--cycle, white);
 filldraw(arc((A + B)/2, A, B)--cycle, gray); 
 filldraw(arc((B + C)/2, A, B)--cycle, white);
\end{asy}
```

If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?

Answer: 210
user: In triangle $ABC$, we have $AB = 7$, $AC = 9$, and $BC = 8$. The angle bisector of $\angle BAC$ intersects side $BC$ at point $D$, and the angle bisector of $\angle ABC$ intersects $AD$ at point $E$. Compute $AE^2$.

Answer: 21
user: Find the number of subsets $S$ of $\{1, 2, \dots, 63\}$ such that the sum of the elements in $S$ is 2008.

Answer: 6
user: An *up-right path* from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_1), \dots, (x_k, y_k)$ of points in $\mathbb{R}^2$ such that $(a, b) = (x_1, y_1)$, $(c, d) = (x_k, y_k)$, and for each $1 \le i < k$, we have that either $(x_{i+1}, y_{i+1}) = (x_i + 1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.\n\nFind the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.

Answer: 1750
user: Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$. Note that the first sequence starts at $99$, and the second sequence ends at $98$.

Answer: 9801
user: Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.

Answer: 7
user: A Mediterranean polynomial has only real roots and is of the form:
\[ P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \]
with real coefficients $a_0, \ldots, a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.

Answer: 11
user: Let $a, b$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}$ and $(a-b)^2 = 4(ab)^3$. Find $\log_a b$.

Answer: -1
user: For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that
\[
\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N
\]
find the sum of the squares of the digits of $N$.

Answer: 56
user: Find the largest positive integer that cannot be expressed in the form $a + bc$, where $a$, $b$, and $c$ are positive integers satisfying $a < b < c$.

Answer: 24
user: Find the number of $4$-digit positive integers $n$ such that the largest power of 2 that divides $n!$ is $2^{n-1}$.

Answer: 4
user: Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?

Answer: 48
user: Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.

Answer: 5792
user: Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?

\begin{asy}
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
\end{asy}

Answer: 42
user: Determine the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.

Answer: 24
user: Find the product of all values of $d$ such that $x^{3} + 2x^{2} + 3x + 4 = 0$ and $x^{2} + dx + 3 = 0$ have a common root.

Answer: 1
user: Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 421
user: Find the minimum possible least common multiple of twenty natural numbers whose sum is $801$.

Answer: 42
user: Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2 + ax + b$ and $x^2 + bx + a$ has two distinct real roots, and the product of these trinomials has exactly three distinct real roots. Determine all possible values of the sum of these three roots.

Answer: 0
user: Let $ABCD$ be an isosceles trapezoid with base $AD$ measuring $11$ cm and the other three sides each measuring $5$ cm. Draw a line perpendicular to $BD$ passing through $C$, and another line perpendicular to $AC$ passing through $B$. Let $E$ be the intersection point of these two lines. Determine the distance between point $E$ and line $AD$.

Answer: 1
user: Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$.

Answer: 210
user: Let $b$ be a real number randomly selected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $\frac{m}{n}$ is the probability that the equation $$x^4 + 25b^2 = (4b^2 - 10b)x^2$$ has \textit{at least} two distinct real solutions. Find the value of $m+n$.

Answer: 63
user: Find all three-digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! \times b! \times c!$.

Answer: 432
user: In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$.

Answer: 40
user: A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be \( \frac{m}{n} \) where \( m, n \) are numbers such that \( m, n \) are positive integers and \( \gcd(m, n) = 1 \).

Take this answer and add the numerator and denominator.

Answer: 3004
user: For a positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$. For example, $f_{1}(123) = (1+2+3)^{2} = 36$. Define $f_{n+1}(k) = f_{1}(f_{n}(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim.

Answer: 169
user: Find the integer represented by $\left\lfloor \sum_{n=1}^{10^9} n^{-2/3} \right\rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

Answer: 2997
user: Georg is putting his $250$ stamps in a new album. On the first page, he places one stamp, and then on every subsequent page, he places either the same number of stamps or twice as many as on the preceding page. In this way, he ends up using precisely all $250$ stamps. How few pages are sufficient for him?

Answer: 8
user: Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\[
\begin{align*}
1+1+1+1&=4,\\
1+3&=4,\\
3+1&=4.
\end{align*}
\]
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$.

Answer: 71
user: The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$.

Answer: 5
user: The function $f$ is defined by the following table:

\[
\begin{array}{|c||c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
f(x) & 4 & 1 & 3 & 5 & 2 \\
\hline
\end{array}
\]

Given $u_0 = 4$ and the recurrence relation $u_{n+1} = f(u_n)$ for $n \geq 0$, find $u_{2002}$.

\[
\textbf{(A) } 1 \quad \textbf{(B) } 2 \quad \textbf{(C) } 3 \quad \textbf{(D) } 4 \quad \textbf{(E) } 5
\]

Answer: 2
user: Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?

Answer: 61
user: A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be $\frac{m}{n}$ where $m,n$ are numbers such that $m,n$ are positive integers and $\gcd(m,n) = 1$. Take this answer and add the numerator and denominator.

Answer: 3004
user: Let $ABCDEF$ be a regular hexagon with $AB = 7$. Let $M$ be the midpoint of $DE$. The lines $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, and $AM$ and $BF$ intersect at $R$. Find the value of $[APB] + [BQC] + [ARF] - [PQMR]$, where $[X]$ denotes the area of polygon $X$.

Answer: 0
user: Find the last three digits of the number $\frac{2019!}{2^{1009}}$.

Answer: 0
user: In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.

Answer: 23
user: Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$

Answer: 0
user: Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2}, \dots, x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that
\[ n = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}. \]

Answer: 130
user: Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?

Answer: 11
user: At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average, how many minutes can a student spend on a problem?

Answer: 16
user: What is the sum of the squares of the roots of the equation $x^2 - 7 \lfloor x \rfloor + 5 = 0$?

Answer: 92
user: Triangle $\triangle ABC$ has $AB= 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $\triangle BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m,n$. Find $m+n$.

Answer: 933
user: The cells of a $2021 \times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?

Answer: 5
user: Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 17
user: Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$, satisfying the condition $p \mid (a^{p-2} - a)$?

Answer: 48
user: Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2 + xy + y^2 = m$ intersects the hyperbola $xy = n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81 \le n \le 81$.

Answer: 284
user: Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation:
\[
LEET + LMT = TOOL
\]
Given that $O = 0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO$.

Answer: 1880
user: Find the sum of the real roots of \(2x^4 + 4x^3 + 6x^2 + 4x - 4\).

Answer: -1
user: What is the sum of all positive integers less than $30$ that are divisible by $2$, $3$, or $5$?

Answer: 315
user: What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?

Answer: 1936
user: We draw a triangle inside a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75^\circ$. 

Next, we draw a second triangle congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference, rotated $75^\circ$ clockwise from the first triangle so that it shares a side with the first triangle. 

We continue this process to draw a third, fourth, and fifth triangle, each rotated $75^\circ$ clockwise from the previous triangle. 

The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by this intersection?

Answer: 120
user: How many trailing zeros does the value \(300 \cdot 305 \cdot 310 \cdots 1090 \cdot 1095 \cdot 1100\) end with?

Answer: 161
user: Compute the number of ordered pairs of integers \((a, b)\), with \(2 \le a, b \le 2021\), that satisfy the equation
\[
a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.
\]

Answer: 43
user: Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the following three congruences:

\[
\begin{aligned}
\left\lfloor \frac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\
\left\lfloor \frac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\
\left\lfloor \frac{n}{256} \right\rfloor &\equiv 1 \pmod{4}.
\end{aligned}
\]

Here, $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Answer: 6491
user: Let $\text{s}_b(k)$ denote the sum of the digits of $k$ in base $b$. Compute \[ \text{s}_{101}(33) + \text{s}_{101}(66) + \text{s}_{101}(99) + \cdots + \text{s}_{101}(3333). \]

Answer: 6683
user: Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive integers.

Answer: 1
user: Find the total number of different integers the function

\[ f(x) = \left\lfloor x \right\rfloor + \left\lfloor 2 \cdot x \right\rfloor + \left\lfloor \frac{5 \cdot x}{3} \right\rfloor + \left\lfloor 3 \cdot x \right\rfloor + \left\lfloor 4 \cdot x \right\rfloor \]

takes for $0 \leq x \leq 100.$

Answer: 734
user: Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.

Answer: 2189
user: Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have:
\[
 (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$.

Answer: 4
user: There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules: 
- If a player holds at least one red card, they pass one red card to the player sitting to their left. 
- Otherwise, they transfer one white card to the left. 

The game ends after the round when each player has one red card and one white card. Determine the maximum number of rounds possible.

Answer: 2009
user: In the given figure, the chord $CD$ is perpendicular to the diameter $AB$ and intersects it at point $H$. The length of $AB$ is a two-digit natural number. By changing the order of these two digits, we obtain the length of $CD$. Knowing that the distance from $H$ to the center $O$ is a positive rational number, calculate the length of $AB$.

Answer: 65
user: A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$.

Answer: 56
user: Find the sum of all integers $n$ that fulfill the equation $$2^n(6-n)=8n.$$

Answer: 9
user: Let $R$ be the rectangle on the Cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$.

Answer: 85
user: Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$.

Answer: 86400
user: In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.

Answer: 3
user: Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]

Among all $n$ and partitions of $S$, determine the minimum possible score.

Answer: 10
user: The quartic (4th-degree) polynomial $P(x)$ satisfies $P(1) = 0$ and attains its maximum value of $3$ at both $x = 2$ and $x = 3$. Compute $P(5)$.

Answer: -24
user: Find the greatest positive integer $N$ with the following property: there exist integers $x_1, \ldots, x_N$ such that $x_i^2 - x_ix_j$ is not divisible by $1111$ for any $i \neq j$.

Answer: 1000
user: Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?

Answer: 108
user: Find the maximum length of a broken line on the surface of a unit cube, such that:
- The line's links are the cube's edges and diagonals of faces.
- The line does not intersect itself.
- The line passes no more than once through any vertex of the cube.
- The endpoints of the line are in two opposite vertices of the cube.

Answer: 7
user: Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

Answer: 540
user: Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks. He notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

Answer: 2517
user: In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent to line $BC$ at point $C$. If $r = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $100m + n$.

Answer: 7508
user: The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon, Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.

Answer: 65
user: Let $BC = 6$, $BX = 3$, $CX = 5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX} \perp \overline{BC}$ and $AF = \sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c + 3b$.

Answer: 1288
user: At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither.

Answer: 271
user: If $x, y, z$ satisfy $x+y+z = 12$, $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$, and $x^3+y^3+z^3 = -480$, find $x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2$.

Answer: 672
user: Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits with a black disc. Then Arutyun comes and identifies both closed digits and their order. What is the minimal $N$ for which they can perform such a trick?

Answer: 101
user: Let $a$ and $b$ be prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

Answer: 10000
user: Determine all 3-digit numbers which are equal to the cube of the sum of all their digits.

Answer: 512
user: Rectangle $ABCD$ has a perimeter of $178$ and an area of $1848$. What is the length of the diagonal of the rectangle?

Answer: 65
user: Let $p(x)$ be a polynomial of degree strictly less than $100$ such that it does not have $(x^3-x)$ as a factor. If \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \) for some polynomials $f(x)$ and $g(x)$, then find the smallest possible degree of $f(x)$.

Answer: 0
user: In a school, there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.

Answer: 23