Incircle of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90$

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

In an acute $ABC$ triangle which has a circumcircle center called $O$, there is a line that perpendiculars to $AO$ line cuts $[AB]$ and $[AC]$ respectively on $D$ and $E$ points. There is a point called $K$ that is different from $AO$ and $BC$'s junction point on $[BC]$. $AK$ line cuts the circumcircle of $ADE$ on $L$ that is different from $A$. $M$ is the symmetry point of $A$ according to $DE$ line. Prove that $K$,$L$,$M$,$O$ are circular.

There are $32$ boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a $15$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? $\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28$

Determine the minimum value of $f (x)$ where f (x) = (3 sin x - 4 cos x - 10)(3 sin x + 4 cos x - 10).

We consider the graph with vertices $A_1,A_2,\dots A_{2015}$ , $B_1,B_2,\dots B_{2015}$ and edges $A_iA _{i+1}, A_iB_i, B_iB_{i+17} $, taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.) Tuymaada 2017 Q8 Juniors

Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

A sequence $\{ u_n \}$ of integers is defined by \[u_1 = 2, u_2 = u_3 = 7,\] \[u_{n+1} = u_nu_{n-1} - u_{n-2}, \text{ for }n \geq 3\] Prove that for each $n \geq 1$, $u_n$ differs by $2$ from an integral square.

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: [list][*]Every row consists of distinct numbers; [*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; [*]$x_{17}=x_{28}=x_{39}=0$; [*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. [*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list] Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.

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 $BB' = CC'$.

A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?

If $b$, $c$ are the legs, and $a$ is the hypotenuse of a right triangle, prove that$$\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 5+3\sqrt{2}$$

Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

Suppose positive real numers $x,y,z,w$ satisfy $(x^3+y^3)^4=z^3+w^3$. Prove that $$x^4z+y^4w\geq zw.$$

Find the degree measure of an angle whose complement is $ 25\%$ of its supplement. $ \textbf{(A)}\ 48 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 150$

Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$. [i]Proposed by Morteza Saghafian - Iran[/i]

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.

At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find $4$ teams , say $A, B, C$ and $D$ so that $A$ defeated $B, C$ and $D, B$ defeated $C$ and $D$ , and $C$ defeated $D$ .

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?