In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

Let $ABC$ be an acute, non-isosceles triangle with $AD$,$BE$, $CF$ are altitudes and $d$ is the tangent line of the circumcircle of triangle $ABC$ at $A$. The line through $H$ and parallel to $EF$ cuts $DE$, $DF$ at $Q, P$ respectively. Prove that $d$ is tangent to the ex-circle respect to vertex $D$ of triangle $DPQ$.

In this problem, a [i]simple polygon[/i] is a polygon that does not intersect itself and has no holes, and a [i]side[/i] of a polygon is a maximal set of collinear, consecutive line segments in the polygon. In particular, we allow two or more consecutive vertices in a simple polygon to be identical, and three or more consecutive vertices in a simple polygon to be collinear. By convention, polygons must have at least three sides. A simple polygon is [i]convex[/i] if every one of its interior angles is $180^\circ$ degrees or less. A simple polygon is concave if it is not [i]convex[/i]. Let P be the plane. Prove or disprove each of the following statements: $(a)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are the vertices of a simple concave $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the vertices of a simple convex polygon in some order (which may or may not have $n$ sides). $(b)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are the vertices of a simple convex $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the vertices of a simple concave polygon in some order (which may or may not have $n$ sides).

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? [asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\displaystyle\frac12$",(0.25,0),S); label("$\displaystyle\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy] $ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $

Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy \[ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, \] and are zero elsewhere.

8. Let $\triangle A B C$ be a triangle with sidelengths $A B=5, B C=7$, and $C A=6$. Let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $L, M, N$ be the midpoints of sides $B C, C A, A B$, respectively. If the area of the convex hexagon with vertices at $D, E, F, L, M, N$ can be written as $\frac{x \sqrt{y}}{z}$ for positive integers $x, y, z$ with $\operatorname{gcd}(x, z)=1$ and $y$ square-free, find $x+y+z$.

Given an acute triangle $ABC$ with $AB < AC$.Let $\Omega $ be the circumcircle of $ ABC$ and $M$ be centeriod of triangle $ABC$.$AH$ is altitude of $ABC$.$MH$ intersect with $\Omega $ at $A'$.prove that circumcircle of triangle $A'HB$ is tangent to $AB$. A.I.Golovanov, A. Yakubov

Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?

The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.

The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.

Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

In the centre of a $2005 \times 2005$ chessboard lies a dice that is to be moved across the board in a sequence of moves. One move consists of the following three steps: - The dice has to be turned with an arbitrary side on top, - then it has to be moved by the shown number of points to the right or left - and finally moved by the concealed number of points upwards or downwards. The attained square is the starting square for the next move. Which squares of the chessboard can be reached in a finite sequence of such moves?

Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Prove for irrational number $\alpha$ and positive integer $n$ that \[ \left( \alpha + \sqrt{\alpha^2 - 1} \right)^{1/n} + \left(\alpha - \sqrt{\alpha^2 - 1} \right)^{1/n} \] is irrational.