ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, where $y < 28$. What is the value of $x + y$?
Answer: 56
Problem: A square has been divided into $2022$ rectangles, with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?
Answer: 2025
Problem: In the diagram, $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^\circ$ and $\angle BOE = 68^\circ$. Find $\angle AOF$.
Answer: 118
Problem: Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy and hops till he falls asleep with $0$ energy. How many different places could he have gone to sleep?
Answer: 10201
Problem: There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:

1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with the three rows (the three cells) all contain $0$.

What is the greatest possible value of $n$?
Answer: 12376
Problem: Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.
Answer: 7
Problem: Find the smallest natural number $k$ for which there exist natural numbers $m$ and $n$ such that $1324 + 279m + 5^n$ is the $k$-th power of some natural number.
Answer: 3
Problem: Let $a, b, c, d$ be the four roots of the polynomial:
\[
x^4 + 3x^3 - x^2 + x - 2.
\]
Given that:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{2}
\]
and
\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} = -\frac{3}{4},
\]
the value of:
\[
\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} + \frac{1}{d^3}
\]
can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\). Compute \(m+n\).
Answer: 39
Problem: Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations:
\[
\begin{align*}
    abcd &= 2007, \\
    a &= \sqrt{55 + \sqrt{k+a}}, \\
    b &= \sqrt{55 - \sqrt{k+b}}, \\
    c &= \sqrt{55 + \sqrt{k-c}}, \\
    d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*}
\]
Answer: 1018
Problem: Determine the value of $p$ such that the numbers $p$, $4p^2 + 1$, and $6p^2 + 1$ are all prime numbers.
Answer: 5
Problem: Two semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line as shown. If $PQ = QR = RS = 24$, find the length of the radius $r$.
Answer: 15
Problem: Jack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the integers 1 through 6 (fair meaning the numbers appear with equal probability). Jack wins if a prime number appears when he rolls, while Jill wins if a number greater than 1 appears when she rolls. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability that Jill wins the game can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 16
Problem: A finite set $M$ of positive integers consists of distinct perfect squares and the number $92$. The average of the numbers in $M$ is $85$. If we remove $92$ from $M$, the average drops to $84$. If $N^2$ is the largest possible square in $M$, what is the value of $N$?
Answer: 22
Problem: Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.
Answer: 0
Problem: Solve in positive integers the following equation:

$$ \left[ \sqrt{1} \right] + \left[ \sqrt{2} \right] + \left[ \sqrt{3} \right] + \ldots + \left[ \sqrt{x^2 - 2} \right] + \left[ \sqrt{x^2 - 1} \right] = 125, $$

where $[a]$ denotes the integer part of the real number $a$.
Answer: 6
Problem: Suppose that $x, y, z$ are non-zero real numbers such that:
\[
\begin{cases}
  x = 2 - \dfrac{y}{z} \\
  y = 2 - \dfrac{z}{x} \\
  z = 2 - \dfrac{x}{y}.
\end{cases}
\]
Find all possible values of $T = x + y + z$.
Answer: 3
Problem: A positive integer $m$ is called perfect if the sum of all its positive divisors, including $1$ and $m$, equals $2m$. Determine the positive integers $n$ for which $n^n + 1$ is a perfect number.
Answer: 3
Problem: Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy:
\[
(ab + 1)(bc + 1)(ca + 1) = 84.
\]
Answer: 6
Problem: On rectangular coordinates, point $A = (1,2)$ and point $B = (3,4)$. Point $P = (a, 0)$ is on the $x$-axis. Given that $P$ is chosen such that the sum $AP + PB$ is minimized, compute $60a$.
Answer: 100
Problem: If integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have?
Answer: 3
Problem: Find the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$.
Answer: 2001
Problem: Given the function $f(x) = x^3 - 6x^2 + 17x$, if $f(a) = 16$ and $f(b) = 20$, find the value of $a + b$.
Answer: 4
Problem: Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$.
Answer: -3
Problem: Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A, B$ and the other at points $C, D$, so that $|AB| = |BC| = |CD| = 14\text{ cm}$. Another line intersects the circles at $E, F$, respectively $G, H$ so that $|EF| = |FG| = |GH| = 6\text{ cm}$. Find the radius $r$.
Answer: 13
Problem: A table tennis club hosts a series of doubles matches following several rules:
1. Each player belongs to two pairs at most.
2. Every two distinct pairs play one game against each other at most.
3. Players in the same pair do not play against each other when they pair with others respectively.

Every player plays a certain number of games in this series. All these distinct numbers make up a set called the "set of games". 

Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding "set of games" is equal to set $A$ exists.
Answer: 6
Problem: Assume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a $90$ degree angle. If each track has a train of length $L$ traveling at the same speed, what is the maximum value of $L$ such that the trains can travel without crashing?
Answer: 15
Problem: On a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a white bucket are selected, and an equal amount of water is added to both. After a certain number of moves, no bucket is empty, and for every pair of buckets that were selected together at least once, the amount of water in these buckets is the same. Find all possible values of $k$.
Answer: 100
Problem: $n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd.
Answer: 5
Problem: To each positive integer $n$, assign a non-negative integer $f(n)$ such that the following conditions are satisfied:

1. $f(rs) = f(r) + f(s)$
2. $f(n) = 0$, if the first digit (from right to left) of $n$ is 3.
3. $f(10) = 0$.

Find $f(1985)$. Justify your answer.
Answer: 0
Problem: Find all the ways of placing the integers $1, 2, 3, \ldots, 16$ in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square. 

\begin{asy}
import graph;
real r=10;
size(r*cm);
picture square1;
draw(square1, (0,0)--(0,1)--(1,1)--(1,0)--cycle);
add(scale(r/31*cm)*square1,(0,0));
picture square2;
draw(square2, (-1,0.5)--(0,0.5)--(0,1)--(1,1)--(1,0)--(0,0)--(0,0.5));
for(int i=1; i<16; ++i)
{
add(scale(r/31*cm)*square2,(i,0));
}
\end{asy}
Answer: 1
Problem: Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.
Answer: 1
Problem: Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least $6$ feet apart from each other?
Answer: 12
Problem: Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$?
Answer: 2
Problem: In an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.
Answer: 129
Problem: Find the largest $k$ such that for every positive integer $n$, there exist at least $k$ numbers in the set \( \{ n+1, n+2, \ldots, n+16 \} \) that are coprime with \( n(n+17) \).
Answer: 1
Problem: For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is $(-1)^{\alpha(i-1, j-1)}$. For $1 \le i, j \le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \le i, j \le 2^{2017}$. Compute the remainder when $\det A$ is divided by $2017$.
Answer: 1382
Problem: What are the last two digits of $2^{3^{4^{\cdots^{2019}}}}$?
Answer: 52
Problem: Define a \textit{subsequence} of a string \( \mathcal{S} \) of letters to be a positive-length string using any number of the letters in \( \mathcal{S} \) in order. For example, a subsequence of \( HARRISON \) is \( ARRON \). Compute the number of subsequences in \( HARRISON \).
Answer: 255
Problem: Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2 - bx + c = 0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.
Answer: 64
Problem: Suppose that $x$, $y$, and $z$ are positive real numbers satisfying the following system of equations:

\[
\begin{cases} 
  x^2 + xy + y^2 = 64 \\
  y^2 + yz + z^2 = 49 \\
  z^2 + zx + x^2 = 57 
\end{cases}
\]

Then, \(\sqrt[3]{xyz}\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Compute \(m + n\).
Answer: 69
Problem: Find the number of ordered pairs of integers \((p, q)\) satisfying the equation \(p^2 - q^2 + p + q = 2014\).
Answer: 16
Problem: We call a pair $(a,b)$ of positive integers a "pupusa" if $a < 391$ and \(\text{lcm}(a,b) > \text{lcm}(a,391)\). Find the minimum value of $b$ across all "pupusa" pairs.
Answer: 18
Problem: Let $x$ be a real number in the interval $(0, \frac{\pi}{2})$ such that $\frac{1}{\sin x \cos x} + 2\cot 2x = \frac{1}{2}$. Evaluate $\frac{1}{\sin x \cos x} - 2\cot 2x$.
Answer: 8
Problem: Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$.
Answer: 2
Problem: Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
\[
[a, b] = 1000, \quad [b, c] = 2000, \quad [c, a] = 2000.
\] 

Where \([x, y]\) represents the least common multiple of positive integers \(x, y\).
Answer: 70
Problem: A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$?
Answer: 250
Problem: Find the smallest multiple of $2019$ that is of the form $abcabc\dots abc$, where $a$, $b$, and $c$ are digits.
Answer: 673673673
Problem: Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
Answer: 111
Problem: For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n} - 1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
Answer: 329
Problem: How many ordered pairs of integers \((x, y)\) are there such that
\[ 0 < \left\vert xy \right\vert < 36? \]
Answer: 524
Problem: How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? 

(Note: A substring is a continuous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)
Answer: 6
Problem: The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ and $y$, erases them, and then writes down $x + y + xy$. He continues this process until only one number is left on the blackboard. What is this number?
Answer: 2010
Problem: Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$, where $m$ and $n$ are nonnegative integers. If $1776$ is one of the numbers that is not expressible, find $a + b$.
Answer: 133
Problem: An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
Answer: 39
Problem: Asheville, Bakersfield, Charter, and Darlington are four small towns along a straight road in that order. The distance from Bakersfield to Charter is \( \frac{1}{3} \) the distance from Asheville to Charter and \( \frac{1}{4} \) the distance from Bakersfield to Darlington. If it is \( 12 \) miles from Bakersfield to Charter, how many miles is it from Asheville to Darlington?
Answer: 72
Problem: What is the sum of all possible values of $\cos(2\theta)$ if $\cos(2\theta) = 2\cos(\theta)$ for a real number $\theta$?
Answer: 2
Problem: Suppose that $ABC$ is a triangle with $AB = 6$, $BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a + b + c$.
Answer: 18
Problem: The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?
Answer: 217
Problem: Nine positive integers $a_1, a_2, \ldots, a_9$ have their last two-digit parts equal to $11, 12, 13, 14, 15, 16, 17, 18,$ and $19$ respectively. Find the last two-digit part of the sum of their squares.
Answer: 85
Problem: Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
Answer: 133
Problem: Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.
Answer: 1763
Problem: Square $A$ is adjacent to square $B$, which is adjacent to square $C$. The three squares all have their bottom sides along a common horizontal line. The upper left vertices of the three squares are collinear. If square $A$ has an area of $24$, and square $B$ has an area of $36$, find the area of square $C$.
Answer: 54
Problem: Let $n \geq 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0 \leq k \leq n-1$ and there exist integers $x, y, z \in \mathbb{Z}$ such that
\[
\begin{align*}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{align*}
\]
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
Answer: 101
Problem: For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\lceil \sqrt{s} \rceil \mid s \in S\}$. Find the number of sets $T$ such that $|f(T)| = 2$ and $f(f(T)) = \{2\}$.
Answer: 5043
Problem: Problem 1: The set $S = \{1, 2, 3, \ldots, 2006\}$ is partitioned into two disjoint subsets $A$ and $B$ such that:

1. $13 \in A$;
2. If $a \in A$, $b \in B$, and $a + b \in S$, then $a + b \in B$;
3. If $a \in A$, $b \in B$, and $ab \in S$, then $ab \in A$.

Determine the number of elements in $A$.
Answer: 154
Problem: It is given a positive real number $a$ such that: 
$$ \left\{ \frac{1}{a} \right\} = \{ a^2 \} $$
$$ 2 < a^2 < 3 $$
Find the value of $a^{12} - \frac{144}{a}$.
Answer: 233
Problem: Find the remainder when $2024^{2023^{2022^{2021 \ldots^{3^{2}}}}} + 2025^{2021^{2017^{2013 \ldots^{5^{1}}}}}$ is divided by $19$.
Answer: 17
Problem: Consider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\sqrt{2}$ and $BC = 10\sqrt{3}$. Construct semicircles $P$, $Q$, and $R$ with diameters $AB$, $AC$, and $BC$ respectively, such that the plane of each semicircle is perpendicular to the plane of $ABC$, and all semicircles are on the same side of plane $ABC$. There exists a plane above triangle $ABC$ that is tangent to all three semicircles $P$, $Q$, and $R$ at the points $D$, $E$, and $F$ respectively. Calculate, with proof, the area of triangle $DEF$.
Answer: 24
Problem: At a round table, there are $10$ students. Each student thinks of a number and communicates this number to their immediate neighbors (left and right) such that others do not hear it. Thus, each student knows three numbers: their own and the two from their neighbors. 

Afterward, each student publicly announces the arithmetic mean of the two numbers they learned from their neighbors. The publicly announced arithmetic means are $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$, respectively. 

Determine the number thought by the student who publicly announced the number $6$.
Answer: 1
Problem: What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?

**Note:** The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left(\frac{a_1^2 + a_2^2 + \cdots + a_n^2}{n}\right)^{1/2}\]
Answer: 337
Problem: A rectangular pool table has vertices at $(0, 0)$, $(12, 0)$, $(0, 10)$, and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.
Answer: 9
Problem: Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.
Answer: 5
Problem: Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides, thus dividing the rectangle into an $m \times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\binom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$.
Answer: 1289
Problem: Find the maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every two different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes immediately after $a$.
Answer: 2014
Problem: Suppose $a, b, c > 0$ are integers such that:
\[
abc - bc - ac - ab + a + b + c = 2013.
\]
Find the number of possibilities for the ordered triple $(a, b, c)$.
Answer: 18
Problem: Find the number of strictly increasing sequences of nonnegative integers with the following properties:

1. The first term is $0$ and the last term is $12$. Therefore, the sequence has at least two terms.
2. Among any two consecutive terms, exactly one of them is even.
Answer: 144
Problem: Let $f(x) = 1 + x + x^2 + \cdots + x^{100}$. Find $f'(1)$.
Answer: 5050
Problem: Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.
Answer: 8
Problem: Let $S = \{ A = (a_1, \ldots, a_s) \mid a_i = 0 \text{ or } 1, \ i = 1, \ldots, 8 \}$. For any two elements of $S$, $A = \{ a_1, \ldots, a_8 \}$ and $B = \{ b_1, \ldots, b_8 \}$. Let $d(A,B) = \sum_{i=1}^{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most, how many elements can $S$ have such that the distance between any two sets is at least 5?
Answer: 4
Problem: One writes 268 numbers around a circle, such that the sum of 20 consecutive numbers is always equal to 75. The numbers 3, 4, and 9 are written in positions 17, 83, and 144, respectively. Find the number in position 210.
Answer: -1
Problem: On a clock, there are two instants between $12$ noon and $1\,\mathrm{PM}$ when the hour hand and the minute hand are at right angles. The difference in minutes between these two instants is written as $a + \dfrac{b}{c}$, where $a, b, c$ are positive integers, with $b < c$ and $\dfrac{b}{c}$ in reduced form. What is the value of $a+b+c$?
Answer: 51
Problem: A sequence of integers $x_1, x_2, \ldots$ is called "double-dipped" if it satisfies the recurrence relation $x_{n+2} = ax_{n+1} + bx_n$ for all $n \ge 1$, where $a$ and $b$ are fixed integers.

Ri starts forming a sequence by randomly picking three integers from the set $\{1, 2, \ldots, 12\}$, with replacement. It is known that if Ri adds a fourth term by picking another element at random from $\{1, 2, \ldots, 12\}$, there is at least a $\frac{1}{3}$ chance that the resulting four-term sequence forms the beginning of a double-dipped sequence.

Given this information, how many distinct three-term sequences could Ri have initially picked?
Answer: 48