ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2 \pmod{n}$. Compute the number of positive integers $k$ such that $k \mid f(k)$.
Answer: 24
Problem: How many ordered pairs of positive integers $(m, n)$ satisfy the system:

\[
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\
\text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
\]

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
Answer: 2
Problem: Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, \( a_i = a_{i-1}^2 - 1 \) if $a_{i-1}$ is odd and \( a_i = \frac{a_{i-1}}{2} \) if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
Answer: 105
Problem: Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?
Answer: 5040
Problem: On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$th, and let $P$ be that perfect square. Find $N + P$.
Answer: 218
Problem: There are $2012$ distinct points in the plane, each of which is to be colored using one of $n$ colors, so that the numbers of points of each color are distinct. A set of $n$ points is said to be multi-colored if their colors are distinct. Determine $n$ that maximizes the number of multi-colored sets.
Answer: 61
Problem: In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational.
Answer: 54
Problem: Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$.
Answer: 50
Problem: How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)
Answer: 12
Problem: A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^2+y^2<1\) in its interior. Compute the largest possible radius of the circle.
Answer: 61
Problem: There are $100$ lightbulbs $B_1, B_2, \ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1, S_2, \ldots, S_{100}$ such that for all $1 \leq i \leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where $B_{101} = B_1$ and $B_0 = B_{100}$). Suppose David chooses whether to flick each switch with probability $\frac{1}{2}$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?
Answer: 50
Problem: A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates the sum of the digits for each card. How many cards does Paulo need to take out of the box to ensure finding at least three cards with the same digit sum?
Answer: 53
Problem: Let $MATH$ be a trapezoid with $MA = AT = TH = 5$ and $MH = 11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.
Answer: 62
Problem: Given a set $I=\{(x_1,x_2,x_3,x_4) \mid x_i \in \{1,2,\cdots,11\}\}$. 

$A \subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4), (y_1,y_2,y_3,y_4) \in A$, there exists $i, j \ (1 \leq i < j \leq 4)$, such that $(x_i - x_j)(y_i - y_j) < 0$. 

Find the maximum value of $|A|$.
Answer: 24
Problem: Positive integers $m$ and $n$ satisfy the equation $$(m + n)(24mn + 1) = 2023.$$ Find the value of $m + n + 12mn$.
Answer: 151
Problem: Sita and Geeta are two sisters. If Sita's age is written after Geeta's age, a four-digit perfect square number is obtained. If the same exercise is repeated after 13 years, another four-digit perfect square number will be obtained. What is the sum of the present ages of Sita and Geeta?
Answer: 55
Problem: How many positive integers $n$, with $n \leq 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n$?
Answer: 1339
Problem: There is a sequence defined by the following conditions:

- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + a\left(\left\lceil \frac{n}{2} \right\rceil \right)$

Find the value of $a(2014)$.

*Note:*
- $\left\lfloor \frac{n}{2} \right\rfloor$ is the floor function, representing the largest integer less than or equal to $\frac{n}{2}$.
- $\left\lceil \frac{n}{2} \right\rceil$ is the ceiling function, representing the smallest integer greater than or equal to $\frac{n}{2}$.
Answer: 34
Problem: Find the number of positive integers less than or equal to $2017$ that have at least one pair of adjacent digits that are both even. For example, count the numbers $24$, $1862$, and $2012$, but not $4$, $58$, or $1276$.
Answer: 738
Problem: In the number arrangement:
\[
\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}
\]
What is the number that will appear directly below the number $2010$?
Answer: 2073
Problem: Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.
Answer: 11
Problem: For every integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and 
\[
(n-k) f_n(k-1) + kf_n(k+1) = nf_n(k)
\] 
for each $1 \le k < n$.  Let 
\[
S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N)
\]
Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$.  (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
Answer: 26
Problem: Let $a$, $b$, $c$ be the roots of the polynomial equation $x^3 - 7x^2 - 6x + 5 = 0$. Compute $(a+b)(a+c)(b+c)$.
Answer: -37
Problem: A semicircle with a radius of $5$ and a quarter circle with a radius of $8$ are tangent to each other and are situated inside a square as shown in the figure. Find the length of the part of the common tangent that is enclosed within the same square. 

![Diagram](https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png)
Answer: 13
Problem: Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$.
Answer: 12
Problem: A quadrilateral is given where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Find the perimeter of this quadrilateral.
Answer: 11
Problem: Consider the $10$-digit number $M = 9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. 

For example, from $M = 9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1} = 9786453210$. Note that any number of (disjoint) pairs can be interchanged. 

Find the number of new numbers that can be so obtained from $M$.
Answer: 88
Problem: Consider an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$. 

Upon the sequence $1, 2, 3, \ldots, 1024$, the operation is performed successively $5$ times. Show that at the end only one number remains and find this number.
Answer: 683
Problem: Let $O$ be the center of the circumcircle of triangle $ABC$. We know that $AB = 1$ and $AO = AC = 2$. Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively, such that $OD = OE$ and $BD = \sqrt{2} \cdot EC$. Find $OD^2$.
Answer: 4
Problem: Determine the greatest natural number $n$ such that for every set $S$ of 2015 different integers, there exist two subsets of $S$ (each subset can have at least one element and the subsets do not need to be disjoint) such that the sum of the elements in each subset is divisible by $n$.
Answer: 2015
Problem: Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.
Answer: 17
Problem: Let $ABC$ be a triangle with $\angle BAC = 117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD \sim \triangle ACB$. Compute the measure of $\angle ABC$, in degrees.
Answer: 42
Problem: In an 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
Problem: Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.
Answer: 20
Problem: The area of the region bound by the equations $y = 2\cos(4x)$, $y = \sin(2x) + 10$, $x=0$, and $x = 5\pi$ can be expressed as $x\pi$ for some value of $x$. Evaluate $x$.
Answer: 50
Problem: The equation
$$ (x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016) $$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Answer: 2016
Problem: Find the natural number $A$ such that there are $A$ integer solutions to $x+y \geq A$ where $0 \leq x \leq 6$ and $0 \leq y \leq 7$.
Answer: 10
Problem: In triangle $ABC$, point $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD$ and $BE$ are perpendicular to $CF$. Let $M$ be the midpoint of $AB$. If $ME = 13$, $AD = 15$, and $BE = 25$, find $AC + CB$.
Answer: 104
Problem: Seongcheol has $3$ red shirts and $2$ green shirts. He cannot distinguish between the red shirts or between the green shirts. In how many ways can he arrange them in a row in his closet such that the two green shirts are not adjacent?
Answer: 6
Problem: Let $\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$?
Answer: 29
Problem: Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.
Answer: 997
Problem: Let $a > 1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n \geq 1}$ is defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct.
Answer: 46
Problem: How many triples \((a,b,c)\) of positive integers strictly less than 51 are there such that \(a+b+c\) is divisible by \(a\), \(b\), and \(c\)?
Answer: 221
Problem: For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $\sum_{n=1}^{210} nf(n)$ is divided by $211$.
Answer: 48
Problem: Hapi, the god of the annual flooding of the Nile, is preparing for this year's flooding. The shape of the channel of the Nile can be described by the function $y = \frac{-1000}{x^2+100}$, where the $x$ and $y$ coordinates are in meters. The current depth of the river is $5$ meters. Hapi plans to increase the water level by $3$ meters. How many meters wide will the river be after the flooding? The depth of the river is always measured at its deepest point.
Answer: 40
Problem: How many positive integers $k$ are there such that $\frac{k}{2013}(a+b) = \text{lcm}(a,b)$ has a solution in positive integers $(a,b)$?
Answer: 1006
Problem: Let $x_1, x_2, \dots, x_{42}$ be real numbers such that $5x_{i+1} - x_i - 3x_ix_{i+1} = 1$ for each $1 \le i \le 42$, with $x_1 = x_{43}$. Find the product of all possible values for $x_1 + x_2 + \dots + x_{42}$.
Answer: 588
Problem: For a positive integer $n$, let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999488$ and $a_7 = 9999999488$. Find the value of $n$ such that $a_n$ is divisible by the highest power of $2$.
Answer: 6
Problem: Determine the least possible value of $f(1998)$, where $f:\mathbb{N} \to \mathbb{N}$ is a function such that for all $m,n \in \mathbb{N}$,  \[ f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}. \]
Answer: 120
Problem: In a $4 \times 4$ table, we place $k$ blocks such that:

1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we remove a block, there is at least one cell that is not covered.

Find the maximum value of $k$.

Note: The blocks can overlap.
Answer: 12
Problem: Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:

\[
1 < a < b+2 < 10.
\]
Answer: 28
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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:

1. $A$ shakes hands with $B$, and $B$ shakes hands with $C$, or
2. $A$ 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.
Answer: 2002
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
Answer: 5
Problem: 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
Problem: 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
Problem: 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