ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Imagine a regular a $2015$-gon with edge length $2$. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle $S$ is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls, $S$ turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the black lengths can be expressed in the form $\tfrac{p\pi}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$. What is $p + q$?
Answer: 2012
Problem: Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form $\tfrac{p}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$, what is $p + q$?
Answer: 8
Problem: Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
Answer: 13
Problem: Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]
Answer: 2
Problem: Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?
Answer: 7
Problem: Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P(1)$?
Answer: 2044
Problem: Let $f(n)$ denote the sum of the distinct positive integer divisors of n. Evaluate:
\[f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9).\]
Answer: 69
Problem: Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder when $N$ is divided by $1000?$
Answer: 382
Problem: Let $P, A, B, C$ be points on circle $O$ such that $C$ does not lie on arc $\widehat{BAP}$, $\overline{P A} = 21, \overline{P B} = 56, \overline{P C} = 35$ and $m \angle BP C = 60^\circ$. Now choose point $D$ on the circle such that $C$ does not lie on arc $\widehat{BDP}$ and $\overline{BD} = 39$. What is $AD$?
Answer: 16
Problem: Let $S$ be the set of ordered integer pairs $(x, y)$ such that $0 < x < y < 42$ and there exists some integer $n$ such that $x^6-y^6 \mid n^2+2015^2$. What is the sum $\sum_{(x_i, y_i) \in S}x_iy_i$?
Answer: 252
Problem: Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?
Answer: 252
Problem: What is the remainder when
\[\sum_{k=0}^{100}10^k\]
is divided by $9$?
Answer: 2
Problem: Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?
Answer: 270
Problem: Every day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s friend remarked that at this rate, it would take Heesu an expected $2800$ days to finally ask Sally out. Suppose $p=\tfrac{m}{n}$, where $\gcd(m, n) = 1$ and $m, n > 0$. What is $m + n$?
Answer: 8
Problem: Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]
Answer: 1847
Problem: Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number n such that $\sigma_1(n) = 1854$?
Answer: 1234
Problem: The lattice points $(i, j)$ for integers $0 \le i, j \le 3$ are each being painted orange or black. Suppose a coloring is good if for every set of integers $x_1, x_2, y_1, y_2$ such that $0 \le x_1 < x_2 \le 3$ and $0 \le y_1 < y_2 \le 3$, the points $(x_1, y_1),(x_1, y_2),(x_2, y_1),(x_2, y_2)$ are not all the same color. How many good colorings are possible?
Answer: 840
Problem: An equilateral triangle $ABC$ has side length $7$. Point $P$ is in the interior of triangle $ABC$, such that $PB=3$ and $PC=5$. The distance between the circumcenters of $ABC$ and $PBC$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime and $m$ and $p$ are relatively prime positive integers. What is $m+n+p$?
Answer: 13
Problem: Let $a\%b$ denote the remainder when $a$ is divided by $b$. Find $\Sigma_{i=1}^{100}(100\%i)$.
Answer: 1701
Problem: In how many different orders can the characters $P  \ U  \ M  \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C?$
Answer: 20
Problem: Suppose $z^{3}=2+2i$, where $i=\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Answer: 3
Problem: Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. (Note that $012$ and $007$ are not $3$-digit numbers.)
Answer: 400185
Problem: The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.
Answer: 42
Problem: Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 9
Problem: Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers $1$ gallon/unit $^2$, and Robert starts at $(1, 0)$. Each second, he moves in a straight line from the point $(\cos(\theta),\sin(\theta))$ to the point $(\cos(\theta+a),\sin(\theta+a))$, where a changes after each movement. a starts out as $253^o$ and decreases by $2^o$ each step. If he takes $89$ steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as $\frac{\sqrt{a}- \sqrt{b}}{c} \cot 1^o$, where $a, b$ and $c$ are positive integers and no prime divisor of $c$ divides both $a$ and $b$ twice. Find $a + b + c$.
Answer: 12
Problem: Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 121
Problem: Define the [i]bigness [/i]of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and [i]bigness [/i]$N$ and another one with integer side lengths and [i]bigness [/i]$N + 1$.
Answer: 55
Problem: The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?
Answer: 90
Problem: Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.
Answer: 37
Problem: Ayase chooses three numbers $a, b, c$ independently and uniformly from the interval $[-1, 1]$.
The probability that $0 < a + b < a < a + b + c$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
Answer: 25
Problem: Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $BC$, and let $X=\Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to $\omega$. Find $YZ^2$.
[i]The following fact may be useful: if $\triangle{ABC}$ has incircle $w$ with incenter $I$ and radius $r$, and $\triangle{DEF}$ is the intouch triangle (i.e. $D$, $E$, $F$ are intersections of incircle with $BC$, $CA$, $AB$, respectively) and $H$ is the orthocenter of $\triangle{DEF}$, then the inversion of $X$ about $\omega$ (i.e. the point $X'$ on ray $IX$ such that $IX' \cdot IX=r^2$) is the midpoint of $DH$.[/i]
Answer: 33
Problem: Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. 
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
Answer: 23
Problem: For a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the [i]Tiger number[/i] of the sequence of flips. How many such sequences have a [i]Tiger number [/i] of $24$?
Answer: 180
Problem: A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$.
Answer: 48
Problem: Let $a \diamond b = ab-4(a+b)+20$. Evaluate
\[1\diamond(2\diamond(3\diamond(\cdots(99\diamond100)\cdots))).\]
Answer: 4
Problem: A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
Answer: 323
Problem: In regular pentagon $ABCDE$, let $O \in CE$ be the center of circle $\Gamma$ tangent to $DA$ and $DE$. $\Gamma$ meets $DE$ at $X$ and $DA$ at $Y$ . Let the altitude from $B$ meet $CD$ at $P$. If $CP = 1$, the area of $\vartriangle COY$ can be written in the form $\frac{a}{b}  \frac{\sin c^o}{\cos^2 c^o}$ , where $a$ and $b$ are relatively prime positive integers and $c$ is an integer in the range $(0, 90)$. Find $a + b + c$.
Answer: 39
Problem: $2017$ voters vote by submitting a ranking of the integers $\{1, 2, ..., 38\}$ from favorite (a vote for that value in $1$st place) to least favorite (a vote for that value in $38$th/last place). Let $a_k$ be the integer that received the most $k$th place votes (the smallest such integer if there is a tie). Find the maximum possible value of $\Sigma_{k=1}^{38}
a_k$.
Answer: 1442
Problem: Compute the number of ordered pairs of integers $(a, b)$, where $0 \le a < 17$ and $0 \le b < 17$, such that $y^2 \equiv x^3 +ax +b \pmod{17}$ has an even number of solutions $(x, y)$, where $0  \le x < 17$ and $0 \le y < 17$ are integers.
Answer: 113
Problem: The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.
Answer: 24
Problem: Define the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$. 
For $n > 1$, if there is a greatest integer $k > 1$ such that $a_n$ is a perfect $k$th power, then $a_{n+1} =\sqrt[k]{a_n}$, otherwise  $a_{n+1} = a_n + b_n$. If $a_{n+1} \ge a_n$ then $b_{n+1} = b_n$, otherwise $b_{n+1} = b_n + 1$. Find $a_{2017}$.
Answer: 8002
Problem: For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le  m \le n$. Find the integer nearest to $b_{36}/b_{25}$.
Answer: 5394
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: [i]Greedy Algorithms, Inc.[/i] offers the following string-processing service. Each string submitted for processing has a starting price of $1$ dollar. The customer can then ask for any two adjacent characters in the string to be swapped. This may be done an arbitrary number of times, but each swap doubles the price for processing the string. Then the company returns the modified string and charges the customer $2^S$ dollars, where $S$ is the number of swaps executed. If a customer submits [b]all [/b]permutations of the string $\text{PUMAC}$ for processing and wants all of the strings to be identical after processing, what is the lowest price, in dollars, she could pay?
Answer: 9765
Problem: Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
Answer: 2
Problem: Let $p(n) = n^4-6n^2-160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
Answer: 7933
Problem: Given a positive integer $x \le  233$, let $a$ be the remainder when $x^{1943}$ is divided by $233$.
Find the sum of all possible values of $a$.
Answer: 932
Problem: The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil  + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree, find $a + b + c$. 
($\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z \rceil$ is the least integer greater than or equal to $z$, for $z \in R$.)
Answer: 30202
Problem: Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$. Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$. What is the area of quadrilateral $ZEBK$?
Answer: 20
Problem: Rectangle $HOMF$ has $HO=11$ and $OM=5$. Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.
Answer: 28
Problem: Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$.
Answer: 2014
Problem: Zack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than $\frac35$ . The expected number of cuts that Zack makes can be written as $\frac{p}{q}$ for $p$ and $q$ relatively prime positive integers. Find $p + q$.
Answer: 34
Problem: Let
\[ \frac{p}{q} = \frac{2017}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\cdots}}}}}}}}}}}\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Answer: 16
Problem: How many ordered pairs of positive integers $(x, y)$ satisfy $yx^y = y^{2017}$?
Answer: 41
Problem: Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$.
Answer: 13446
Problem: Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.
Answer: 680
Problem: If $N$ is the number of ways to place $16$ [i]jumping [/i]rooks on an $8 \times 8$ chessboard such that each rook attacks exactly two other rooks, find the remainder when $N$ is divided by $1000$. (A jumping rook is said to [i]attack [/i]a square if the square is in the same row or in the same column as the rook.)
Answer: 840
Problem: Equilateral triangle $ABC$ has area $1$. $A'$, $B'$, and $C'$ are the midpoints of $BC$, $CA$, and $AB$, respectively. $A''$, $B''$, $C''$ are the midpoints of $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be written as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Find $m+n$.
Answer: 21
Problem: There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 5
Problem: Call a positive integer $n$ [i]tubular [/i] if for any two distinct primes $p$ and $q$ dividing $n, (p + q) | n$. Find the number of tubular numbers less than $100,000$. (Integer powers of primes, including $1, 3$, and $16$, are not considered [i]tubular[/i].)
Answer: 26
Problem: There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 1009
Problem: Let $X =\{1, 2, ... , 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1\le  r \le k$, there exist $k$ subsets of $X$ such that the union of any $ r$ of them is equal to $X$ , but the union of any fewer than $r$ of them is not equal to $X$ . Find, with proof, the greatest possible value for $k$.
Answer: 13
Problem: Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime. 
Find the number of feral triples where $1  \le a < b < c \le 20$.
Answer: 72
Problem: Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.
Answer: 19
Problem: Let $a_n$ be the least positive integer the sum of whose digits is $n$. Find $a_1 + a_2 + a_3 + \dots + a_{20}$.
Answer: 1074
Problem: In the country of Princetonia, there are an infinite number of cities, connected by roads. For every two distinct cities, there is a unique sequence of roads that leads from one city to the other. Moreover, there are exactly three roads from every city. On a sunny morning in early July, n tourists have arrived at the capital of Princetonia. They repeat the following process every day: in every city that contains three or more tourists, three tourists are picked and one moves to each of the three cities connected to the original one by roads. If there are $2$ or fewer tourists in the city, they do nothing. After some time, all tourists will settle and there will be no more changing cities. For how many values of n from $1$ to $2020$ will the tourists end in a configuration in which no two of them are in the same city?
Answer: 19
Problem: Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $AD$, $BE$, $CF$ all contain a point $K$. Let $L$ be the second intersection of the circumcircles of $BFK$ and $CEK$. Suppose that $\frac{AK}{KD} = \frac{11}{7}$ , and $BD = 6$. If $KL^2 =\frac{a}{b}$, where $a, b$ are relatively prime integers, find $a + b$.
Answer: 497
Problem: Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?
Answer: 202