A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. [i]L. Emelyanov, A. Polyansky[/i]

Find all real solutions $(x,y)$ of the system $x^2+y=12=y^2+x$.

Each edge of a complete graph with $101$ vertices is marked with $1$ or $-1$. It is known that absolute value of the sum of numbers on all the edges is less than $150$. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero. [i](Y. Caro, A. Hansberg, J. Lauri, C. Zarb)[/i]

Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

Determine all solutions to the system of equations \begin{align*}x+y+z=2 \\ x^2-y^2-z^2=2 \\ x-3y^2+z=0\end{align*}

A list of $2018$ numbers is created using the following procedure: the first number is $47$, the second number is $74$, and from there, each number is equal to the number formed by the last two digits of the sum of the two previous numbers:$$47, 74, 21, 95, 16, 11, \dots$$ Bruno squares each of the $2018$ numbers and sums them. Determine the remainder when this sum is divided by $8$.

Let $ABC$ be a triangle with circumcircle $\Omega$, circumcenter $O$ and orthocenter $H$. Let $S$ lie on $\Omega$ and $P$ lie on $BC$ such that $\angle ASP=90^\circ$, line $SH$ intersects the circumcircle of $\triangle APS$ at $X\neq S$. Suppose $OP$ intersects $CA,AB$ at $Q,R$, respectively, $QY,RZ$ are the altitude of $\triangle AQR$. Prove that $X,Y,Z$ are collinear. [i]Proposed by Shuang-Yen Lee[/i]

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is: ${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $

In an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

Let $ ABCD$ be a parallelogram with $ \angle BAD \equal{} 60$ and denote by $ E$ the intersection of its diagonals. The circumcircle of triangle $ ACD$ meets the line $ BA$ at $ K \ne A$, the line $ BD$ at $ P \ne D$ and the line $ BC$ at $ L\ne C$. The line $ EP$ intersects the circumcircle of triangle $ CEL$ at points $ E$ and $ M$. Prove that triangles $ KLM$ and $ CAP$ are congruent.

Is it possible to cover a $2003 \times 2003$ chessboard (without overlap) using only horizontal $1 \times 2$ dominoes and only vertical $3 \times 1$ trominoes?

Let $X$ be an arbitrary point on side $BC$ of triangle ABC. Triangle $T$ formed by the bisectors of the angles $ABC$, $ACB$ and $AXC$. Prove that: a) the circumscribed circle of the triangle $T$ passes through the vertex $A$. b) the orthocenter of triangle $T$ lies on line $BC$. (Dmytro Prokopenko)

Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs $+, -, \times$ and a special sign $\pm$. Usage of $\pm$ is equivalent to usage of $+$ and $-$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6\}$, while $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3\}$. Can Peter construct an expression if the numbers on the blackboard are : (a) $1, 2, 4$ ? [i]($2$ points)[/i] (b) any $100$ distinct real numbers ? [i]($6$ points)[/i]

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $[\frac{a}{b}, \frac{c}{d} )$ where $a,b,c,d$ are positive integers with $\gcd(a,b) = \gcd(c,d) = 1$. Compute $1000a+100b+10c+d$.

Let $B, C, D, E$ be four pairwise distinct collinear points and let $A$ be a point not on ine $BC$. Now, let the circumcircle of $\triangle ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$. Show that $DEFG$ is cyclic if and only if $AB=AC$.

Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^{5}$, no three of which are consecutive terms of an arithmetic progression?

Signals used for communication are binary sequences of length $10$. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?

Let $a,b,c$ be three positive real numbers such that $a + b +c =1$ . prove that among the three numbers $a-ab, b - bc, c-ca$ there is one which is at most $\frac{1}{4}$ and there is one which is at least $\frac{2}{9}$.

Inside a square with sides of length $1$ unit several non-overlapping smaller squares with sides parallel to the sides of the large square are placed (the small squares may differ in size). Draw a diagonal of the large square and consider all of the small squares intersecting it. Can the sum of their perimeters be greater than $1993$? (AN Vblmogorov)

Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?