Mark the Martian and Bark the Bartian live on planet Blok, in the year $2019$. Mark and Bark decide to play a game on a $10 \times 10$ grid of cells. First, Mark randomly generates a subset $S$ of $\{1, 2, \dots, 2019\}$ with $|S|=100$. Then, Bark writes each of the $100$ integers in a different cell of the $10 \times 10$ grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is $n$, then she stacks $n$ $1 \times 1 \times 1$ blocks on top of one other in that cell. Let $B$ be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the $100$ integers into the grid of cells. Find the expected value of $B$ over all possible sets $S$ Mark could have generated. [i]Proposed by Yang Liu[/i]

For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?

The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: $ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 72 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 36$

Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon. (V Proizvolov)

Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies \[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \] Prove that $\lim_{n\to \infty}c_n$ exists and find its value. [i]Proposed by Sadovnichy-Grigorian-Konyagin[/i]

Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$. [color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]

We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.

Suppose that circles $C_1$ and $C_2$ intersect at $X$ and $Y$ . Let $A, B$ be on $C_1$, $C_2$, respectively, such that $A, X, B$ lie on a line in that order. Let $A, C$ be on $C_1$, $C_2$, respectively, such that $A, Y, C$ lie on a line in that order. Let $A', B', C'$ be another similarly defined triangle with $A \ne A'$. Prove that $BB' = CC'$.

Let $n$ be a positive integer. Let us call a sequence $a_1,a_2,\dots,a_n$ interesting if for any $i=1,2,\dots,n$ either $a_i=i$ or $a_i=i+1$. Let us call an interesting sequence even if the sum of its members is even, and odd otherwise. Alice has multiplied all numbers in each odd interesting sequence and has written the result in her notebook. Bob, in his notebook, has done the same for each even interesting sequence. In which notebook is the sum of the numbers greater than by how much? (The answer may depend on $n$.)

Define sequence $\{a_n\}$: $a_1$ is any positive integer, and for any positive integer $n\ge 1$, $a_{n+1}$ is the smallest positive integer coprime to $\sum_{i=1}^{n} a_i$ and not equal to $a_1,\ldots, a_n$. Prove that every positive integer appears in the sequence $\{a_n\}$.

Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$. (Peter Novotný)

We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

All faces of the tetrahedron $ABCD $ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$

A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

Solve the following in equation in $\mathbb{R}^3$: \[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]

Let $RANDOM$ be a regular hexagon with side length $1.$ Points $I$ and $T$ lie on segments $\overline{RA}$ and $\overline{DO},$ respectively, such that $MI=MT$ and $\angle{TMI}=90^\circ.$ Compute the area of triangle $MIT.$

The set of all finite ordered sets of $0$ and $ 1$ is somehow partitioned into two disjoint classes. Prove that any infinite sequence of $0$ and $1$ can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.