The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.

Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with \[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\] Find $ \det A$.

Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.

Prove that the equation $$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$ has $2013$ integer solutions.

There are \(n\) numbers arranged in a circle, and each number equals the absolute value of the difference between its two neighbors. Is it necessarily true that all numbers are equal to zero if: a) \(n=2025\); b) \(n=2024\)? [i]Proposed by Anton Trygub[/i]

It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]

Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions? [i]A. Golovanov[/i]

Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.

We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge. Prove that for all integers $n\geq 3$ there exists a face of the polyhedra with a number of sides not divisible by $n$.

Triangle $ABC$ has incircle $\omega$ and incenter $I$. On its sides $AB$ and $BC$ we pick points $P$ and $Q$ respectively, so that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\omega$ in $T$. Draw a tangent line to $\omega$ passing through $T$; it intersects the sides $AB$ and $BC$ in $U$ and $V$ respectively. Prove that $PU = UV + VQ$!

There exists 4 circles, $ a,b,c,d$, such that $ a$ is tangent to both $ b$ and $ d$, $ b$ is tangent to both $ a$ and $ c$, $ c$ is both tangent to $ b$ and $ d$, and $ d$ is both tangent to $ a$ and $ c$. Show that all these tangent points are located on a circle.

The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?

In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of  $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?

Several tiles congruent to the one shown in the picture below are to be fit inside a $11 \times 11$ square table, with each tile covering $6$ whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. [img]https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png[/img] Poland

Two triangles are said to be [i]twins [/i] if one of them is an image of the other one under a parallel projection. Prove that two triangles are twins if and only if either at least a side of one of them equals a side of another or both the triangles have equal segments that connect the corresponding vertices with some points on the opposite sides which divide these sides in the same ratio. (E. Barabanov)

Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$? [asy] draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--cycle); draw((0, 0)--(sqrt(3)/3, 1)); draw((0, 0)--(sqrt(3), 1)); label("A", (0, 1), N); label("B", (2, 1), N); label("C", (2, 0), S); label("D", (0, 0), S); label("E", (sqrt(3)/3, 1), N); label("F", (sqrt(3), 1), N); [/asy] ${ \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$

Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.

Let $H$ be an arbitrary point on the altitude $CP$ of the acute triangle $ABC$. The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$, respectively. [list] (a) Prove that $\angle NPC =\angle MPC$. (b) Let $O$ be the common point of $MN$ and $CP$. An arbitrary line through $O$ meets the sides of quadrilateral $CNHM$ in $D$ and $E$. Prove that $\angle EPC =\angle DPC$. [/list]