Determine the smallest integer $n \ge 2012$ for which it is possible to have $16$ consecutive integers on a $4 \times 4$ board so that, if we select $4$ elements of which there are not two in the same row or in the same column, the sum of them is always equal to $n$. For the $n$ found, show how to fill the board.

$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$.

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

Write $0.2010\overline{228}$ as a fraction.

There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several 16-person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other fifteen people. What is the minimum number of meetings needed until every person knows everyone else's name? [i]David Yang, Victor Wang.[/i] [size=85][i]See the "odd version" [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500914]here[/url].[/i][/size]

For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent? (translated by Miklós Maróti)

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$. [i]Proposed by Andrew Wen[/i]

For $x;y \geq 0$ prove the inequality: $\sqrt{x^2-x+1} \sqrt{y^2-y+1}+ \sqrt{x^2+x+1} \sqrt{y^2+y+1} \geq 2(x+y)$

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Given a circle $k$ and the point $P$ outside it, an arbitrary line $s$ passing through $P$ intersects $k$ at the points $A$ and $B$ . Let $M$ and $N$ be the midpoints of the arcs determined by the points $A$ and $B$ and let $C$ be the point on $AB$ such that $PC^2=PA\cdot PB$ . Prove that $\angle MCN$ doesn't depend on the choice of $s$. [color=red][Moderator edit: This problem has also been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=56295 .][/color]

A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. [img]https://cdn.artofproblemsolving.com/attachments/b/5/ba11377252c278c4154a8c3257faf363430ef7.png[/img] Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.

Given a rectangle $EFGH$ with $EF=3$ and $FG=2010,$ mark a point $P$ on $FG$ such that $FP=4.$ A laser beam is shot from $E$ to $P,$ which then reflects off $FG,$ then $EH,$ then $FG,$ etc. Once it reaches some point of $GH,$ the beam is absorbed; it stops reflecting. How far does the beam travel?

Let $\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$. Let points $K,L$ be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$. Prove that the points $B,K,C,L$ are concyclic if and only if $\angle BAC=90^{\circ}$.

Let $P(x)$ be a polynomial with integer coefficients such that \[P(\sqrt{2}\sin x) = -P(\sqrt{2}\cos x)\] for all real numbers $x$. What is the largest prime that must divide $P(2019)$?

Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.

For positive integers $j\le n$, prove that $$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$ [i]Proposed by Ángel Plaza[/i]

Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$. Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square.

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$. [i]Proposed by Márton Lovas, Budapest[/i]

(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle. a) All three segments are equal. Is it true that the triangle is equilateral? b) Two segments are equal. Is it true that the triangle is isosceles? c) Can the segments have length 4, 4 and 3?

Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$

In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move: [i]They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$.[/i] John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

If $a+3b = 9$ and $a+11b =21$, what is the missing coefficient in the expression $2a+\underline{?}b = 27$? $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 9\qquad\textbf{(D) } 12\qquad\textbf{(E) } 14$