We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles are there?

Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.

Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. [i]Proposed by Navid.[/i]

In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre $I$ touches the side BC at D. (The circle also touches the sides of $AB$, $AC$ extended.) Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.

Let $a_1, a_2, \dots, a_{2^{2016}}$ be positive integers not bigger than $2016$. We know that for each $n \leq 2^{2016}$, $a_1a_2 \dots a_{n} +1 $ is a perfect square. Prove that for some $i $ , $a_i=1$.

Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? $\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$

Let $H$ and $O$ be the orthocenter and circumcenter of an acute-angled triangle $ABC$, respectively. The perpendicular bisector of $BH$ meets $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. Prove that $OB$ bisects the angle $A_1OC_1$.

For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$ (S. Manukian, Yerevan)

There are $n>1$ cities in Jansonland, with two-way roads joining certain pairs of cities. Janson will send a few robots one-by-one to build more roads. The robots operate as such: 1. Janson first selects an integer $k$ and a list of cities $a_0, a_1, \dots, a_k$ (cities can repeat). 2. The robot begins at $a_0$ and goes to $a_1$, then $a_2$, and so on until $a_k$. 3. When the robot goes from $a_i$ to $a_{i+1}$, if there is no road then the robot builds a road, but if there is a road then the robot destroys the road. In terms of $n$, determine the smallest constant $k$ such that Janson can always achieve a configuration such that every pair of cities has a road connecting them using no more than $k$ robots. [i](Proposed by Ho Janson)[/i]

Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)

Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

Prove that there are $100$ distinct positive integers $n_1,n_2,\dots,n_{99},n_{100}$ such that $\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}$ is a perfect cube.

A tennis league has three teams, and each team plays the each of the other two teams twice. How many total matches are there, between these three tennis teams? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }12$

The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full? [asy] draw((5,0)..(0,-1.3)..(-5,0)); draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); draw(ellipse((0,10),5,1.3)); draw(circle((.3,1.3),.4)); draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle);[/asy] $ \text{(A)}\ 80\qquad\text{(B)}\ 100\qquad\text{(C)}\ 125\qquad\text{(D)}\ 130\qquad\text{(E)}\ 262 $

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

A valid lottery ticket is formed by choosing $5$ distinct numbers from $1, 2,3,..., 90$. What is the probability that the winning ticket contains at least two consecutive numbers?

$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

Homer gives mathematicians Patty and Selma each a different integer, not known to the other or to you. Homer tells them, within each other’s hearing, that the number given to Patty is the product $ab$ of the positive integers $a$ and $b$, and that the number given to Selma is the sum $a + b$ of the same numbers $a$ and $b$, where $b > a > 1.$ He doesn’t, however, tell Patty or Selma the numbers $a$ and $b.$ The following (honest) conversation then takes place: Patty: “I can’t tell what numbers $a$ and $b$ are.” Selma: “I knew before that you couldn’t tell.” Patty: “In that case, I now know what $a$ and $b$ are.” Selma: “Now I also know what $a$ and $b$ are.” Supposing that Homer tells you (but neither Patty nor Selma) that neither $a$ nor $b$ is greater than 20, find $a$ and $b$, and prove your answer can result in the conversation above.

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. [img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]