A right hexagon with side length $n$ is divided into tiles of three types, which are shown in the image, which are rhombuses with side length $1$ each and the acute angle $60$. In one move you can choose three tiles, arranged as shown in the image on the left, and rearrange them, as shown in the image on the right [img]https://iili.io/dxEvyqN.jpg[/img] Moves are made until it is impossible to make a move. a) Prove that for the fixed initial arrangement of tiles the same amount of moves would be made independent of the moves. b) For each positive integer $n$ find the maximum number of moves among all possible initial arrangements [i]M. Zorka[/i]

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular. [i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.

Decide whether or not there exists a nonconstant polynomial $Q(x)$ with integer coefficients with the following property: for every positive integer $n > 2$, the numbers \[ Q(0), \; Q(1), Q(2), \; \dots, \; Q(n-1) \] produce at most $0.499n$ distinct residues when taken modulo $n$. [i]Proposed by Yang Liu[/i]

Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.

Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.

$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$. Prove $bc \geq b+c$

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

Write down one of the following integers: $1, 2, 4, 8, 16.$ If your team is the only one that submits this integer, you will receive that number of points; otherwise, you receive zero. [b][color=#f00]There's no real way to solve this but during the competition, each of the 5 available scores were submitted at least twice by the 16 teams competing. [/color][/b]

Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.

Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\ x + y + 2z = 4(a^2 + 1) \\ z^2 - xy = a^2 \end{cases}$

We are given a 7x7 square board. In each square, one of the diagonals is traced, and then one of the two formed triangles is colored blue. What is the largest area a continuous blue component can have? (Note: continuous blue component means a set of blue triangles connected via their edges, passing through corners is not permitted)

Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?

Two positive integers $m$ and $n$ satisfy $$max \,(m, n) = (m - n)^2$$ $$gcd \,(m, n) = \frac{min \,(m, n)}{6}$$ Find $lcm\,(m, n)$

The diagram below shows a large square divided into nine congruent smaller squares. There are circles inscribed in five of the smaller squares. The total area covered by all the five circles is $20\pi$. Find the area of the large square. [asy] size(80); defaultpen(linewidth(0.6)); pair cent[] = {(0,0),(0,2),(1,1),(2,0),(2,2)}; for(int i=0;i<=3;++i) { draw((0,i)--(3,i)); } for(int j=0;j<=3;++j) { draw((j,0)--(j,3)); } for(int k=0;k<=4;++k) { draw(circle((cent[k].x+.5,cent[k].y+.5),.5)); } [/asy]

How many subsets of $\{1, 2,...,9\}$ do not contain $2$ adjacent numbers?

In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$. [i]Proposed by Liam Baker, South Africa[/i]

[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $ [b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that $$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$ for any nonnegative integer $ n. $

Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.

Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be the feet of the perpendiculars drawn from point $A$ on the lines $A_1D$ and $A_1C$, respectively, and the points $P$ and $Q$ be the feet of the perpendiculars drawn from point $B_1$ on the lines $A_1C_1$ and $A_1C$, respectively. Prove that $\angle EFA = \angle PQB_1$

A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number?

FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$