Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

When we write the number $n > 2$ as the sum of some integers consecutive positives (at least two addends), we say that we have an [i]elegant decomposition[/i] of $n$. Two [i]elegant decompositions[/i] will be different if any of them contains some term that does not contains the other. How many different elegant decompositions does the number $3^{2004}$ have?

Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

Find all triples of positive integers $(x,y,z)$ that satisfy the equation $$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

We define two types of operation on polynomial of third degree: a) switch places of the coefficients of polynomial(including zero coefficients), ex: $ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$ b) replace the polynomial $P(x)$ with $P(x+1)$ If limitless amount of operations is allowed, is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?

Kevin develops a method for shuffling a stack of $10$ cards numbered $1$ through $10$. He starts with the unshuffled pile, which is in perfect order with $1$ at the top and $10$ at the bottom. He takes the top card off the unshuffled pile and places it in what he calls the shuffled pile. Then, he flips a coin. If the coin is heads, he takes the card at the top of the unshuffled pile and places it at the top of the shuffled pile. If the coin comes up tails, he places the card at the at the bottom of the shuffled pile. He repeats this process for all the remaining cards. What is the probability that at the end of this shuffling, the top card is a prime number? Express your answer as a common fraction.

Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions? $ \textbf{(A) }\text{6}\qquad\textbf{(B) }\text{9}\qquad\textbf{(C) }\text{12}\qquad\textbf{(D) }\text{18}\qquad\textbf{(E) }\text{24} $

The circles $k_1$ and $k_2$ are tangent at the point $P$. A line is drawn through $P$, cutting $k_1$ at $A_1$ and $k_2$ at $A_2$. A second line is drawn through $P$, cutting $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that the triangles $PA_1B_1$ and $PA_2B_2$ are similar.

The graph of the function $ f(x) \equal{} 2x^3 \minus{} 7$ goes: $ \textbf{(A)}\ \text{up to the right and down to the left} \\ \textbf{(B)}\ \text{down to the right and up to the left} \\ \textbf{(C)}\ \text{up to the right and up to the left} \\ \textbf{(D)}\ \text{down to the right and down to the left} \\ \textbf{(E)}\ \text{none of these ways.}$

Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps. Find those pair of integers $(k,n)$ for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

How many $5 \times 5$ grids are possible such that each element is either 1 or 0 and each row sum and column sum is $4 ?$ [list=1] [*] 64 [*] 32 [*] 120 [*] 96 [/list]

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers. b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.

Construct a square, if you know its center and two points that lie on adjacent sides.

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \[a_1 + k, a_2 + 2k, \dots, a_p + pk\] produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$. [i]Proposed by Ankan Bhattacharya[/i]

Given a triangle $ABC$ with $\angle C = 90^o$. a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle. b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

A non-equilateral triangle $ABC$ is inscribed in a circle $\Gamma$ with centre $O$, radius $R$ and its incircle has centre $I$ and touches $BC,CA,AB$ at $D,E, F$, respectively. A circle with centre $I$ and radius $r$ intersects the rays $[ID), [IE), [IF)$ at $A',B',C'$. Show that the orthocentre $K$ of $\vartriangle A'B'C'$ is on the line $OI$ and that $\frac{IK}{IO}=\frac{r}{R}$ Michel Bataille .

Consider $5$ distinct positive integers. Can their mean be a)Exactly $3$ times larger than their largest common divisor? b)Exactly $2$ times larger than their largest common divisor?

Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.