ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A positive integer is considered **happy** if:

1. All its digits are different and not $0$.
2. One of its digits is equal to the sum of the other digits.

For example, 253 is a **happy** number. How many **happy** numbers are there?
Answer: 264
Problem: A number is called \textit{6-composite} if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is \textit{composite} if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
Answer: 441
Problem: Suppose that $20^{21} = 2^a 5^b = 4^c 5^d = 8^e 5^f$ for positive integers $a, b, c, d, e,$ and $f$. Find $\frac{100bdf}{ace}$.
Answer: 75
Problem: Define the two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations:
- $a_{n+1} = 3a_n + b_n$
- $b_{n+1} = 3b_n - a_n$

for all nonnegative integers $n$. Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31$, respectively. Compute $100r + s$.
Answer: 3010
Problem: The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. Given that $P(3) = 89$, determine the value of $P(10)$.
Answer: 859
Problem: Let $f$ be a quadratic function that satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$.

For two distinct real numbers $a$ and $b$, if $f(a) = f(b)$, then $f(a^2 - 6b - 1) = f(b^2 + 8)$.
Answer: 13
Problem: Determine the sum of the positive integers $n$ such that there exist primes $p, q, r$ satisfying $p^n + q^2 = r^2$.
Answer: 5
Problem: Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$.
Answer: 1012
Problem: The four sets $A$, $B$, $C$, and $D$ each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets?
Answer: 1094
Problem: For a natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \leq n$, define an $n,s,k$-block as a string of $k$ consecutive elements from $s$. We say that two $n,s,k$-blocks, namely, $a_1a_2\ldots a_k$ and $b_1b_2\ldots b_k$, are incompatible if there exists an $i \in \{1,2,\ldots ,k\}$ such that $a_i \neq b_i$. Additionally, for two natural numbers $r \leq n$ and $l$, we say that $s$ is $r,l$-typed if there are at most $l$ pairwise incompatible $n,s,r$-blocks. 

Let $t$ be a $3,7$-typed string consisting of $10,000$ binary digits. Determine the maximum number $M$ such that $t$ is $10,M$-typed.
Answer: 128
Problem: Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.
Answer: 80
Problem: Determine all even numbers $n$, $n \in \mathbb{N}$ such that:
\[
\frac{1}{d_{1}} + \frac{1}{d_{2}} + \cdots + \frac{1}{d_{k}} = \frac{1620}{1003},
\]
where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.
Answer: 2006
Problem: Find all prime numbers of the form $\frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}$, where $n$ is a natural number.
Answer: 101
Problem: Find the sum of all the positive integers that are divisors of either $96$ or $180$.
Answer: 770
Problem: Find the largest positive integer $n$ ($n \ge 3$), such that there exists a convex $n$-gon where the tangent of each interior angle is an integer.
Answer: 8
Problem: Find the smallest positive integer $n$ such that $\sqrt{\frac{1^2 + 2^2 + \ldots + n^2}{n}}$ is an integer.
Answer: 337
Problem: Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are:

- Two black cards
- Two red cards
- One black and one red card

Determine the probability that Hong and Song play exactly the same pair of cards in each of the four turns. This probability is given as $\frac{m}{n}$, where $m$ and $n$ are positive integers such that $\gcd(m, n) = 1$. Calculate $100m + n$.
Answer: 25150
Problem: Suppose that $n$ is a positive integer and that $a$ is the integer equal to \( \frac{10^{2n}-1}{3(10^n+1)} \). If the sum of the digits of $a$ is 567, what is the value of $n$?
Answer: 189
Problem: Compute the number of digits in $2015!$.
Answer: 5786
Problem: A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$.
Answer: 1257
Problem: Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a > b > c > d$ are positive integers satisfying:

$$|\zeta^a + \zeta^b + \zeta^c + \zeta^d| = \sqrt{3}.$$  

Compute the smallest possible value of $1000a + 100b + 10c + d$.
Answer: 7521
Problem: How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer root?
Answer: 49
Problem: Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions:

- $f(1) = 2$
- $f(a) + f(b) \leq 2\sqrt{f(a)}$
- For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b + 100)$.
Answer: 10201
Problem: There are $360$ permutations of the letters in $MMATHS$. When ordered alphabetically, starting from $AHMMST$, $MMATHS$ is in the $n$th permutation. What is $n$?
Answer: 173
Problem: Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a \neq 20, 22$. Find the minimum possible positive value of $P(0)$.
Answer: 20
Problem: In the triangle $ABC$, $\angle ABC = \angle ACB = 78^\circ$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^\circ$, $\angle CBE = 51^\circ$. Find the measure of angle $\angle BED$.
Answer: 12
Problem: Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$, where $n$ and $p$ are squarefree, find $m+n+p$.
Answer: 11
Problem: Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2 + b^2 + c^2 + a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m \rfloor$.
Answer: 594
Problem: What are the last two digits of the number $$(11 + 12 + 13 + \ldots + 2006)^2$$?
Answer: 56
Problem: In $\triangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Draw the circumcircle of $\triangle ABC$, and suppose that the circumcircle has center $O$. Extend $AO$ past $O$ to a point $D$, $BO$ past $O$ to a point $E$, and $CO$ past $O$ to a point $F$ such that $D, E, F$ also lie on the circumcircle. Compute the area of the hexagon $AFBDCE$.
Answer: 168
Problem: Compute the number of sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold:

1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$.
2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}$ such that $a_i + a_j - N$ is divisible by $203$.
Answer: 20503
Problem: The repeating decimal $2.0151515\ldots$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 199
Problem: What is the minimal number of operations needed to repaint an entirely white grid $100 \times 100$ to be entirely black, if in one move we can choose $99$ cells from any row or column and change their color?
Answer: 200
Problem: Compute the number of integer solutions $(x, y)$ to the equation $xy - 18x - 35y = 1890$.
Answer: 96
Problem: Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?
Answer: 840
Problem: If $m$, $n$, and $p$ are three different natural numbers, each between $2$ and $9$, what are all the possible integer values of the expression $\frac{m+n+p}{m+n}$?
Answer: 2
Problem: Pick out three numbers from $0, 1, \ldots, 9$, such that their sum is an even number and not less than $10$. We have ________ different ways to pick these numbers.
Answer: 51
Problem: What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits is the same when read backwards.)
Answer: 21
Problem: For each $n \in \mathbb{N}$, let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find the value of $d_1 + d_2 + \cdots + d_{2018} + d_{2019}$.
Answer: 6725
Problem: Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with every possible height-weight combination.

The USAMTS wants to field a competitive team, so there are some strict requirements:

- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team.

Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?
Answer: 128
Problem: The intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.
Answer: 25
Problem: A board measuring $64$ inches in length and $4$ inches in height is inclined such that the longer side of the board forms a $30^\circ$ angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is the value of $a + b + c$?
Answer: 37
Problem: In the figure below, each side of the rhombus has a length of 5 centimeters. 

\[ \text{[asy]} 
  \text{import graph;} 
  \text{unitsize}(2.0\text{cm}); 
  \text{real } w = \sqrt{3}; 
  \text{draw}((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- \text{cycle}); 
  \text{filldraw}(
    \text{Circle}((-0.5, 0), \frac{0.8}{\sqrt{\pi}}), \text{gray}
  ); 
  \text{label}("$60^\circ$", (w - 0.1, 0), \text{W}); 
\]

The circle lies entirely within the rhombus. The area of the circle is \( n \) square centimeters, where \( n \) is a positive integer. Compute the number of possible values of \( n \).
Answer: 14
Problem: Find the sum of all real numbers $x$ such that \( x^2 = 5x + 6\sqrt{x} - 3 \).
Answer: 7
Problem: What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times \ldots \times 50$ is divisible by $10^n$?
Answer: 11
Problem: We call $\triangle ABC$ with centroid $G$ "balanced" on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B$. $\triangle XYZ$, with $XY=2023$ and $\angle ZXY=120^\circ$, is balanced on $XY$. What is the maximum value of $XZ$?
Answer: 4046
Problem: Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$, $C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $\triangle AJG$ is an equilateral triangle, the area of $\triangle AJG$ is $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 48
Problem: A positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$.
Answer: 7
Problem: Calculate the least integer greater than $5^{(-6)(-5)(-4)...(2)(3)(4)}$.
Answer: 2
Problem: Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
Answer: 22200
Problem: Ryan is altering Brice’s coin so that one side appears twice as often as the other. He randomly decides to weight either heads or tails more heavily, with equal probability.

Brice flips this modified coin twice, and it lands on heads both times. We need to find the probability that the coin lands on heads in the next flip. This probability can be expressed as $\frac{p}{q}$, where $p$ and $q$ are positive integers and $\gcd(p, q) = 1$. Determine the value of $p + q$.
Answer: 8
Problem: Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.
Answer: 420
Problem: Let $a$ and $b$ be integer solutions to $17a + 6b = 13$. What is the smallest possible positive value for $a - b$?
Answer: 17
Problem: You have a $10 \times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$, $\ldots$, $10$ from left to right across the top row, then $11$, $12$, $\ldots$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$, $\ldots$, $10$ in the first column (from top to bottom), then $11$, $12$, $\ldots$, $20$ in the second column, and so forth. 

When this process is finished, how many squares will have the property that their two numbers sum to $101$?
Answer: 10
Problem: Compute the number of nonempty subsets $S \subseteq \{-10, -9, -8, \ldots, 8, 9, 10\}$ that satisfy $|S| + \min(S) \cdot \max(S) = 0$.
Answer: 335
Problem: There are $1988$ birds in $994$ cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?
Answer: 1987
Problem: Let $N$ be the number of ordered pairs of integers $(x, y)$ such that \[ 4x^2 + 9y^2 \le 1000000000. \] Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$?
Answer: 52
Problem: On a 30-question test, Question 1 is worth 1 point, Question 2 is worth 2 points, and so on up to Question 30. David takes the test and afterward finds out he answered nine of the questions incorrectly. However, he was not told which nine were incorrect. What is the highest possible score he could have attained?
Answer: 420
Problem: Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
Answer: 84
Problem: Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively. If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
Answer: 180
Problem: Given the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer. Additionally, the following conditions are satisfied:
- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$

Determine the value of $a^b + c^d + e$.
Answer: 628
Problem: A village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is located outside the village, precisely $16 \, \mathrm{m}$ north of the north gate. This tree is visible on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. Determine the diameter, in meters, of the wall that encircles the village.
Answer: 48
Problem: A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.
Answer: 224
Problem: Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.
Answer: 0
Problem: Equilateral triangle $\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\triangle BFD$ and $\triangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $\gcd(a, c) = 1$ and $b$ squarefree, find $a + b + c$.
Answer: 14
Problem: Find the number of positive integers $k \leq 2018$ for which there exist integers $m$ and $n$ such that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Answer: 65
Problem: Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx - 3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x) = 0$ and $g(x) = 0$ have a common root?
Answer: 0
Problem: Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
Answer: -2
Problem: Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
Answer: 90720
Problem: Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital \textit{every} month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes. Beginning in October, the committee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that \textit{must} go by (including October when the first set of five teachers is selected, but not September) such that some five-member committee \textit{must have} taught together twice (all five members are the same during two different months)?
Answer: 1261
Problem: In the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs "The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.

Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$.

Let $N$ be the smallest positive integer so that 
\[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]
is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$.
Answer: 1040
Problem: Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.
Answer: 880
Problem: In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac{2}{3}$. Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.
Answer: 8
Problem: Recall that the Fibonacci numbers $F(n)$ are defined by:
- $F(0) = 0$
- $F(1) = 1$
- $F(n) = F(n - 1) + F(n - 2)$ for $n \geq 2$.

Determine the last digit of $F(2006)$.
Answer: 3
Problem: During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour, he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $\frac{n^2}{2}$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?
Answer: 580
Problem: Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide \( \frac{L}{2016} \).
Answer: 44
Problem: Determine the number of real solutions $a$ to the equation:
\[ \left\lfloor \frac{1}{2}a \right\rfloor + \left\lfloor \frac{1}{3}a \right\rfloor + \left\lfloor \frac{1}{5}a \right\rfloor = a. \]
Here, if $x$ is a real number, then $\left\lfloor x \right\rfloor$ denotes the greatest integer that is less than or equal to $x$.
Answer: 30
Problem: Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$.
Answer: 48
Problem: One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc$, where each of $a$, $b$, and $c$ is a digit between $0$ and $9$, inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the markers will be given a unique two-digit identification number $xy$, with each of $x$ and $y$ a digit between $0$ and $9$, inclusive. Marker $xy$ will be able to mark any contest with an identification number of the form $xyA$, $xAy$, or $Axy$, for any digit $A$. What is the minimum possible number of markers to be hired to ensure that all contests will be marked?
Answer: 50
Problem: Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$
Answer: 673