A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

[u]Round 1[/u] [b]p1.[/b] What is the smallest number equal to its cube? [b]p2.[/b] Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequence of colors are identical) [b]p3.[/b] Jocelyn labels the three corners of a triangle with three consecutive natural numbers. She then labels each edge with the sum of the two numbers on the vertices it touches, and labels the center with the sum of all three edges. If the total sum of all labels on her triangle is $120$, what is the value of the smallest label? [u]Round 2[/u] [b]p4.[/b] Adam cooks a pie in the shape of a regular hexagon with side length $12$, and wants to cut it into right triangular pieces with angles $30^o$, $60^o$, and $90^o$, each with shortest side $3$. What is the maximum number of such pieces he can make? [b]p5.[/b] If $f(x) =\frac{1}{2-x}$ and $g(x) = 1-\frac{1}{x}$ , what is the value of $f(g(f(g(... f(g(f(2019))) ...))))$, where there are $2019$ functions total, counting both $f$ and $g$? [b]p6.[/b] Fhomas is buying spaghetti again, which is only sold in two types of boxes: a $200$ gram box and a $500$ gram box, each with a fixed price. If Fhomas wants to buy exactly $800$ grams, he must spend $\$8:80$, but if he wants to buy exactly 900 grams, he only needs to spend $\$7:90$! In dollars, how much more does the $500$ gram box cost than the $200$ gram box? [u]Round 3[/u] [b]p7.[/b] Given that $$\begin{cases} a + 5b + 9c = 1 \\ 4a + 2b + 3c = 2 \\ 7a + 8b + 6c = 9\end{cases}$$ what is $741a + 825b + 639c$? [b]p8.[/b] Hexagon $JAMESU$ has line of symmetry $MU$ (i.e., quadrilaterals $JAMU$ and $SEMU$ are reflections of each other), and $JA = AM = ME = ES = 1$. If all angles of $JAMESU$ are $135$ degrees except for right angles at $A$ and $E$, find the length of side $US$. [b]p9.[/b] Max is parked at the $11$ mile mark on a highway, when his pet cheetah, Min, leaps out of the car and starts running up the highway at its maximum speed. At the same time, Max starts his car and starts driving down the highway at $\frac12$ his maximum speed, driving all the way to the $10$ mile mark before realizing that his cheetah is gone! Max then immediately reverses directions and starts driving back up the highway at his maximum speed, nally catching up to Min at the $20$ mile mark. What is the ratio between Max's max speed and Min's max speed? [u]Round 4[/u] [b]p10.[/b] Kevin owns three non-adjacent square plots of land, each with side length an integer number of meters, whose total area is $2019$ m$^2$. What is the minimum sum of the perimeters of his three plots, in meters? [b]p11.[/b] Given a $5\times 5$ array of lattice points, how many squares are there with vertices all lying on these points? [b]p12.[/b] Let right triangle $ABC$ have $\angle A = 90^o$, $AB = 6$, and $AC = 8$. Let points $D,E$ be on side $AC$ such that $AD = EC = 2$, and let points $F,G$ be on side $BC$ such that $BF = FG = 3$. Find the area of quadrilateral $FGED$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949413p26408203]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every positive integer $n \geq 3$ there exist two sets $A =\{ x_1, x_2,\ldots, x_n\}$ and $B =\{ y_1, y_2,\ldots, y_n\}$ for which [b]i)[/b] $A \cap B = \varnothing.$ [b]ii)[/b] $x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.$ [b]ii)[/b] $x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.$

Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.

Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $\lambda$ be a positive real number satisfying $\lambda=\lambda^{2/3}+1$. Show that there exists a positive integer $M$ such that $|M-\lambda^{300}|<4^{-100}$. [i]Proposed by Evan Chen[/i]

Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.

Given a positive integer $k$. Prove that there are infinitely many positive integers $n$ satisfy the following conditions at the same time: a) $n$ has at least $k$ distinct prime divisors b) All prime divisors other than $3$ of $n$ have the form $4t+1$, with $t$ some positive integer. c) $n | 2^{\sigma(n)}-1$ Here $\sigma(n)$ demotes the sum of the positive integer divisors of $n$.

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

Prove that the number $a=2019^{2020}+4^{2019}$ is a composite number.

Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2$.

For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.

Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?

Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$

Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which $$ ({\rm gcd}(a_i,a_j))^2|a_k. $$

For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?

Show that there is no infinite sequence of primes $p_1, p_2, p_3, . . .$ there any for each $ k$: $p_{k+1} = 2p_k - 1$ or $p_{k+1} = 2p_k + 1$ is fulfilled. Note that not the same formula for every $k$.

Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]

Let $x$ > $1$ be an integer. Prove that $x^5$ + $x$ + $1$ is divisible by at least two distinct prime numbers.

a) $101$ numbers are selected from the set $1, 2, . . . , 200$. Prove that among the numbers selected there is a pair in which one number is divisible by the other. b) One number less than $16$, and $99$ other numbers are selected from the set $1, 2, . . . , 200$. Prove that among the selected numbers there are two such that one divides the other.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]