The bisectors $BB_1$ and $CC_1$ of the acute triangle $ABC$ intersect in point $I$. On the extensions of the segments $BB_1$ and $CC_1$, the points $B'$ and $C'$ are marked, respectively So, the quadrilateral $AB'IC'$ is a parallelogram. Prove that if $\angle BAC = 60^o$, then the straight line $B'C'$ passes through the intersection point of the circumscribed circles of the triangles $BC_1B'$ and $CB_1C'$.

Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins? $\textbf{(A)} \dfrac{7}{576} \qquad \textbf{(B)} \dfrac{5}{192} \qquad \textbf{(C)} \dfrac{1}{36} \qquad \textbf{(D)} \dfrac{5}{144} \qquad \textbf{(E)}\dfrac{7}{48}$

For any positive numbers $ a_1 , a_2 , \dots, a_n $ prove the inequality $$\! \left(\!1\!+\!\frac{1}{a_1(1+a_1)} \!\right)\! \left(\!1\!+\!\frac{1}{a_2(1+a_2)} \! \right) \! \dots \! \left(\!1\!+\!\frac{1}{a_n(1+a_n)} \! \right) \geq \left(\!1\!+\!\frac{1}{p(1+p)} \! \right)^{\! n} \! ,$$ where $p=\sqrt[n]{a_1 a_2 \dots a_n}$.

Let $k > 2$ be a real number. a) Prove that for all positive real numbers $x,y$ and $z$ the following inequality holds: $$\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ b) Prove that there exist positive real numbers $x, y$ and $z$ such that $$\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ Leonard Giugiuc

On an $8 \times 8$ chessboard, a rook stands on the bottom left corner square. We want to move it to the upper right corner, subject to the following rules: we have to move the rook exactly $9$ times, such that the length of each move is either $3$ or $4$. (It is allowed to mix the two lengths throughout the "journey".) How many ways are there to do this? In each move, the rook moves horizontally or vertically.

Let $ABC$ be a triangle right-angled at $A$ and $S$ be its circumcircle. Let $S_1$ be the circle touching the lines $AB$ and $AC$, and the circle $S$ internally. Further, let $S_2$ be the circle touching the lines $AB$ and $AC$ and the circle $S$ externally. If $r_1, r_2$ be the radii of $S_1, S_2$ prove that $r_1 \cdot r_2 = 4 A[ABC]$.

Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.

$f$ is a function from the set of positive integers to the set of all integers that satisfies the following. [b]$\cdot$[/b] $f(1)=1, f(2)=-1$ [b]$\cdot$[/b] $f(n)+f(n+1)+f(n+2)=f(\left\lfloor\frac{n+2}{3}\right\rfloor)$ Find the number of positive integers $k$ not exceeding $1000$ such that $f(3)+f(6)+\cdots+f(3k-3)+f(3k)=5$.

Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that $W(k,2)=\Omega (2^{\frac{k}{2}})$.

Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: \[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\] Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

You are playing a game called "Hovse." Initially you have the number $0$ on a blackboard. If at any moment the number $x$ is written on the board, you can either: $\bullet$ replace $x$ with $3x + 1$ $\bullet$ replace $x$ with $9x + 1$ $\bullet$ replace $x$ with $27x + 3$ $\bullet$ or replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor $. However, you are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"

Let $ABC$ be a triangle with circumcircle $\omega$. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B \parallel l_C$. The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B \parallel OP \parallel l_C$. [i]Proposed by Stefan Lozanovski, Macedonia[/i]

Let \(\Omega\) be a circle with a chord \(AB\) which is not a diameter. \(\Gamma_{1}\) be a circle on one side of \(AB\) such that it is tangent to \(AB\) at \(C\) and internally tangent to \(\Omega\) at \(D\). Likewise, let \(\Gamma_{2}\) be a circle on the other side of \(AB\) such that it is tangent to \(AB\) at \(E\) and internally tangent to \(\Omega\) at \(F\). Suppose the line \(DC\) intersects \(\Omega\) at \(X \neq D\) and the line \(FE\) intersects \(\Omega\) at \(Y \neq F\). Prove that \(XY\) is a diameter of \(\Omega\) .

Given a circle and two points $P$ and $Q$, construct a right triangle inscribed in the circle such that its two legs pass through the points $P$ and $Q$ respectively. For what positions of $P$ and $Q$ is this construction impossible?

Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a [i]quad[/i]. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a [i]bullet[/i]. Suppose some of the bullets are coloured red. For each pair $(i j)$, with $ 1 \le i < j \le 8$, let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$.

A survey of participants was conducted at the Olympiad. $50\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $70\%$ of the participants liked the opening of the Olympiad. It is known that each participant liked either one option or all three. Determine the percentage of participants who rated all three events positively.

Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.

Given $\triangle ABC$ with $AB < AC$, let $\omega$ be the circle centered at the midpoint $M$ of $BC$ with diameter $AC - AB$. The internal bisector of $\angle BAC$ intersects $\omega$ at distinct points $X$ and $Y$. Let $T$ be the point on the plane such that $TX$ and $TY$ are tangent to $\omega$. Prove that $AT$ is perpendicular to $BC$.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ such that for any colouring of the cells of a $2n\times k$ table with $n$ colours there are two rows and two columns which intersect in four squares of the same colour.

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation?

Suppose 51 numbers are chosen from 1, 2, 3, ..., 99, 100. Show that there are two such that one divides the other.

Find all twice continuously differentiable functions $f: \mathbb{R} \to (0, \infty)$ satisfying $f''(x)f(x) \ge 2f'(x)^2.$