$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

Determine the last three digits of \[2003^{2002^{2001}}.\]

Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC. (The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.) [i]Proposed by Ivan Borsenco[/i]

What is the last three digits of $49^{303}\cdot 3993^{202}\cdot 39^{606}$? $ \textbf{(A)}\ 001 \qquad\textbf{(B)}\ 081 \qquad\textbf{(C)}\ 561 \qquad\textbf{(D)}\ 721 \qquad\textbf{(E)}\ 961 $

Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$ [i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers. [i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$

(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A,B,\cdots,Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within $20$ years of each other, then your score is $0$. Otherwise, your score is $\max(3(k-3),0)$ where $k$ is the number of elements in your set. \[\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}\]

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

Three numbers are initially written on the board: 2023, 2024, and 2025. In each move, you can increase any two of these numbers by 1 and decrease the third one by 2. a) Determine whether it is possible to perform a sequence of operations such that the board eventually contains two numbers that are equal. b) Calculate the number of all possible ordered triples of positive integers that can be obtained by performing such operations some number of times. [i]Proposed by Giorgi Arabidze, Georgia [/i]

A group of $6$ friends sit in the back row of an otherwise empty movie theater. Each row in the theater contains $8$ seats. Euler and Gauss are best friends, so they must sit next to each other, with no empty seat between them. However, Lagrange called them names at lunch, so he cannot sit in an adjacent seat to either Euler or Gauss. In how many different ways can the $6$ friends be seated in the back row? $\text{(A) }2520\qquad\text{(B) }3600\qquad\text{(C) }4080\qquad\text{(D) }5040\qquad\text{(E) }7200$

Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that: \[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}. \]

Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. Determine the ratio $\frac{AK}{KD}$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell. [i]Proposed by Giorgi Arabidze, Georgia[/i]

$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.

Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]