Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

There are given a triangle and some internal point $P$. $x,y,z$ are distances from $P$ to the vertices $A,B$ and $C$. $p,q,r$ are distances from $P$ to the sides $BC,CA,AB$ respectively. Prove that: $$xyz\ge(q+r)(r+p)(p+q).$$

For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$.

How many zeroes are there at the end the number $9^{999} + 1$?

Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.

Given a triangle $ABC$ such that the circumcenter is in the interior of the incircle, prove that the triangle $ABC$ is acute-angled.

Prove that if a quadratic equation $$ ax^2 + bx + c = 0$$ with integer coefficients has a rational root, then at least one of the numbers $ a $, $ b $, $ c $ is even.

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

An integer $n $ is called silesian if there exist positive integers $a,b$ and $c$ such that $$n=\frac{a^2+b^2+c^2}{ab+bc+ca}.$$ $(a)$ prove that there are infinitely many silesian integers. $(b)$ prove that not every positive integer is silesian.

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.

What is the remainder of $2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))$ when it is divided by $2023$? Here $\wedge$ is the exponential symbol, for example $2\wedge (3\wedge 2) = 2\wedge 9 = 512$. The power tower contains the integers from $2025$ to $1$ exactly once, except that the number $2023$ is missing.

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task? $ \textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM} $

Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $ P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.

Assume that $ A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $ l$ taken arbitrarily on the plane, assume that projections of $ A_1, A_2, \ldots, A_8$ on the line are $ P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $ P_{i_1}, P_{i_2}, \ldots, P_{i_8}$ according to the direction of line $ l.$ Thus we get one permutation for $ 1, 2, \ldots, 8,$ namely, $ i_1, i_2, \ldots, i_8.$ In the figure, this permutation is $ 2, 1, 8, 3, 7, 4, 6, 5.$ Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $ N_8 \equal{} N(A_1, A_2, \ldots, A_8).$ Find the maximal value of $ N_8.$

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

You are given $n \geq 3$ distinct real numbers. Prove that one can choose either $3$ numbers with positive sum, or $2$ numbers with negative sum. [i]Proposed by Mykhailo Shtandenko[/i]

For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]

Let $ n \ge 2$ and $x_1$, $x_2$, $...$, $x_n$ be real numbers from the segment $[1,\sqrt2]$. Prove that holds the inequality $$\frac{\sqrt{x_1^2-1}}{x_2}+\frac{\sqrt{x_2^2-1}}{x_3}+...+\frac{\sqrt{x_n^2-1}}{x_1} \le \frac{\sqrt2}{2} n.$$

Viorel participates in a mathematics competition with 50 problems. For each problem he answers correctly, he earns 4 points, and for each problem he answers incorrectly, he loses 1 point. If Viorel answered every problem and has 65 points, how many problems did he solve correctly?

Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.

If $\zeta$ denote the Riemann Zeta Function, and $s>1$ then $$\sum \limits_{k=1}^{\infty} \frac{1}{1+k^s}\geq \frac{\zeta (s)}{1+\zeta (s)}$$

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.