Prove that for all positive real numbers $a,b,c$ the following inequality takes place \[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \] [i]Laurentiu Panaitopol, Romania[/i]

Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.

Let $x, y$ be real numbers such that $x-y, x^2-y^2, x^3-y^3$ are all prime numbers. Prove that $x-y=3$. EDIT: Problem submitted by Leonel Castillo, Panama.

An convex qudrilateral $ABCD$ is given. $O$ is the intersection point of the diagonals $AC$ and $BD$. The points $A_1,B_1,C_1, D_1$ lie respectively on $AO, BO, CO, DO$ such that $AA_1=CC_1, BB_1=DD_1$. The circumcircles of $\triangle AOB$ and $\triangle COD$ meet at second time at $M$ and the the circumcircles of $\triangle AOD$ and $\triangle BOC$ - at $N$. The circumcircles of $\triangle A_1OB_1$ and $\triangle C_1OD_1$ meet at second time at $P$ and the the circumcircles of $\triangle A_1OD_1$ and $\triangle B_1OC_1$ - at $Q$. Prove that the quadrilateral $MNPQ$ is cyclic.

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

Determine all natural numbers $ \overline{ab} $ with $ a<b $ which are equal with the sum of all the natural numbers between $ a $ and $ b, $ inclusively.

In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible. Prove that they didn't get any coin with the value $12$ Kulotnik.

Find all pairs of real numbers $(x, y)$ for which the inequality $y^2 + y + \sqrt{y - x^2 -xy} \le 3xy$ holds.

Starting with an arbitrary pair (a,b) of vectors on the plane, we are allowed to perform the operations of the following two types: (1) To replace $(a,b)$ with $(a+2kb,b)$ for an arbitrary integer $k \ne 0$; (2) To replace $(a,b)$ with $(a,b+2ka)$ for an arbitrary integer $ k \ne 0$. However, we must change the type of operetion in any step. (a) Is it possible to obtain $((1,0), (2,1))$ from $((1,0), (0,1))$, if the first operation is of the type (1)? (b) Find all pairs of vectors that can be obtained from $((1,0), (0,1))$ (the type of first operation can be selected arbitrarily).

A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains: [list]$f(7) = 77$ $f(b) = 85$, where $b$ is Beth's age, $f(c) = 0$, where $c$ is Charles' age.[/list] How old is each child?

Just as the twins finish their masterpiece of symbol art, Wendy comes along. Wendy is impressed by the explanation Alexis and Joshua give her as to how they knew they drew every row exactly once. Wendy puts them both to the test. "Suppose the two of you draw symbols as you have before, stars in pairs and boxes in threes." Wendy continues, "Now, suppose that I draw circles with X's in the middle." Wendy shows them examples of such rows: \[\begin{array}{ccccccccccccccc} \vspace{10pt}*&*&*&*&\otimes&*&*&\otimes&*&*&*&*&\blacksquare&\blacksquare&\blacksquare\\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&*&*&*&*&\otimes&*&*&\otimes&*&*&*&*\\\vspace{10pt}\otimes&\blacksquare&\blacksquare&\blacksquare&\otimes&\otimes&*&*&\otimes&*&*&\otimes&\blacksquare&\blacksquare&\blacksquare \end{array}\] "Again we count both the first two rows, which are mirror images of one another, but we only count a row that is its own mirror image. $\textit{Now}$ how man rows of $15$ symbols are possible?" Though it takes the twins some time, they eventually come up with an answer they agree on. Wendy confirms that they are correct. How many rows did the twins find are possible using all three symbols as described?

Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying $$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$

Point $M$ is the midpoint of side $BC$ of convex quadrilateral $ABCD$. If $\angle{AMD} < 120^{\circ}$. Prove that $$(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD.$$

5. Alice and Bob have found 100 bricks of the same size, 50 white and 50 black. They came up with the following game. A tower will mean one or several bricks standing on top of one another. At the start of the game all bricks lie separately, so there are 100 towers. In a single turn a player must put one of the towers on top of another tower (no flipping towers allowed) so that the resulting tower has no same-colored bricks next to each other. The players make moves in turns, Alice starts first. The one unable to make the next move loses the game. Who can guarantee the win regardless of the opponent's strategy?

Alice is playing a game with $2018$ boxes, numbered $1 - 2018$, and a number of balls. At the beginning, boxes $1 - 2017$ have one ball each, and box $2018$ has $2018n$ balls. Every turn, Alice chooses $i$ and $j$ with $i > j$, and moves exactly $i$ balls from box $i$ to box $j$. Alice wins if all balls end up in box $1$. What is the minimum value of n so that Alice can win this game?

Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $ Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $

We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?

The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience. [list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$. [b](b)[/b] Determine all $n$ for which this is possible.[/list]

A collection of $n > 2$ numbers is called [i]crowded [/i] if each of them is less than their sum divided by $n - 1$ . Let $\{a, b, c, ,...\}$ be a crowded collection of $n$ numbers whose sum equals $S$. Prove that: (a) each of the numbers is positive, (b) we always have $a + b > c$, (c) we always have $a + b \ge \frac{S}{n-1}$ . (Regina Schleifer)

We are given a $\Delta ABC$. Point $D$ on the circumscribed circle k is such that $CD$ is a symmedian in $\Delta ABC$. Let $X$ and $Y$ be on the rays $\overrightarrow{CB}$ and $\overrightarrow{CA}$, so that $CX=2CA$ and $CY=2CB$. Prove that the circle, tangent externally to $k$ and to the lines $CA$ and $CB$, is tangent to the circumscribed circle of $\Delta XDY$.

Let $a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}$. a) Prove that the sequences $(a_{n})$ and $(b_{n})$ are decreasing and converge to $0$. b) Prove that the sequence $(2^{n}a_{n})$ is increasing, the sequence $(2^{n}b_{n})$ is decreasing and both converge to the same limit. c) Prove that there exists a positive constant $C$ such that for all $n$ the following inequality holds: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.

Consider a square ABCD. A point P was selected on its diagonal AC. Let H be the orthocenter of the triangle APD, let M be the midpoint of AD and N be the midpoint of CD. Prove that PN is orthogonal to MH.

Let $S$ be a set which is closed under the binary operation $\circ$, with the following properties: (i) there is an element $e \in S$ such that $a \circ e = e \circ a = a$, for each $a \in S$. (ii) $(a \circ b) \circ (c \circ d)=(a \circ c) \circ (b \circ d)$, for all $a,b, c,d \in S$. Prove or disprove: (a) $\circ$ is associative on S (b) $\circ$ is commutative on S

$BC$ is a segment, $M$ is point on $BC$, $A$ is a point such that $A, B, C$ are non-collinear. (i) Prove that if $M$ is the midpoint of $BC$, then $AB^2 + AC^2 = 2(AM^2 + BM^2).$ (ii) If there exists another point (except $M$) on segment $BC$ satisfying (i), find the region of point $A$ might occupy.

$99$ different points $P_1, P_2, ..., P_{99}$ are marked on circle $O$. For each $P_i$, define $n_i$ as the number of marked points you encounter starting from $P_i$ to its antipode, moving clockwise. Prove the following inequality. $$n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900$$