Suppose that $a,b,c,d$ are non-negative real numbers such that $a^2+b^2+c^2+d^2=2$ and $ab+bc+cd+da=1$. Find the maximum value of $a+b+c+d$ and determine all equality cases.

Let $ ABC$ be a triangle with $ \angle BAC \equal{} 90^\circ$. A circle is tangent to the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, such that the points on the circle diametrically opposite $ X$ and $ Y$ both lie on the side $ BC$. Given that $ AB \equal{} 6$, find the area of the portion of the circle that lies outside the triangle. [asy]import olympiad; import math; import graph; unitsize(20mm); defaultpen(fontsize(8pt)); pair A = (0,0); pair B = A + right; pair C = A + up; pair O = (1/3, 1/3); pair Xprime = (1/3,2/3); pair Yprime = (2/3,1/3); fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white); draw(A--B--C--cycle); draw(Circle(O, 1/3)); draw((0,1/3)--(2/3,1/3)); draw((1/3,0)--(1/3,2/3)); label("$A$",A, SW); label("$B$",B, down); label("$C$",C, left); label("$X$",(1/3,0), down); label("$Y$",(0,1/3), left);[/asy]

Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.

Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.

A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there originally were girls. How many boys and girls were there on board at the start?

On a board, Ana and Bob start writing $0$s and $1$s alternatively until each of them has written $2021$ digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones. Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.

The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that \[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\] where $S_X$ denotes the surface of figure $X$. [i]Proposed by Morteza Saghafian, Ali khezeli[/i]

A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)

The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$

The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$

Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: (1)$a_1=1$ (2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$ what is the residue when we divide $f(2015)$ by $4$ ?

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] All the positive integers from 1 till 1000 are written on the blackboard in some order and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1\le a_1<a_2<\ldots<a_{10}\le1000$ in collection then it is allowed to arrange in increasing order the numbers at places $a_1, a_2, \ldots, a_{10},$ counting from left to right. What is the smallest amount of cards in the collection which enables us to arrange in increasing order all the numbers for any initial arrangement of them?

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.

There are 21 competitors with distinct skill levels numbered 1, 2,..., 21. They participate in a ping-pong tournament as follows. First, a random competitor is chosen to be "active", while the rest are "inactive." Every round, a random inactive competitor is chosen to play against the current active one. The player with the higher skill will win and become (or remain) active, while the loser will be eliminated from the tournament. The tournament lasts for 20 rounds, after which there will only be one player remaining. Alice is the competitor with skill 11. What is the expected number of games that she will get to play?

Each face of a regular tetrahedron with edge length $2011$ is divided into $2011^2$ equilateral triangles of side length $1$, created by drawing lines parallel to each edge. Bruno and Mariano take turns marking one of the unit triangles. Except for the first move, every triangle marked must share at least one point with the triangle marked in the previous move. Bruno plays first. The game ends when a player cannot make a move, and that player loses. Determine which of the two players has a winning strategy and describe the strategy.

For a given positive integer $n$, you may perform a series of steps. At each step, you may apply an operation: you may increase your number by one, or if your number is divisible by 2, you may divide your number by 2. Let $\ell(n)$ be the minimum number of operations needed to transform the number $n$ to 1 (for example, $\ell(1) = 0$ and $\ell(7) = 4$). How many positive integers $n$ are there such that $\ell(n) \leq 12$?

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_{2}$ be radius of $(AEF)$. If $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$ where $a,b,c$ are integers. Find $a^{b^{c}}$

$f(x)=ax^2+bx+c;a,b,c$ are reals. $M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\}$ Prove that $M=N$ or $M \cap N = \O $

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.

Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$. Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$, respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$, respectively. Let $S$ be the circumcenter of triangle $A'B'C'$, and let $H'$ be the orthocenter of triangle $DEF$. Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$. Prove that $HJ$ is parallel to $H'S$.