Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $

The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$. What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$?

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $$\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$? $\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}$

Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board? Clarification: Two squares are [i]neighbors [/i] if they have a common side.

a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

Given a convex quadrilateral $ABCD$, in which $\angle CBD = 90^o$, $\angle BCD =\angle CAD$ and $AD= 2BC$. Prove that $CA =CD$. (Anton Trygub)

Convex circumscribed quadrilateral $ABCD$ with its incenter $I$ is given such that its incircle is tangent to $\overline{AD},\overline{DC},\overline{CB},$ and $\overline{BA}$ at $K,L,M,$ and $N$. Lines $\overline{AD}$ and $\overline{BC}$ meet at $E$ and lines $\overline{AB}$ and $\overline{CD}$ meet at $F$. Let $\overline{KM}$ intersects $\overline{AB}$ and $\overline{CD}$ at $X,Y$, respectively. Let $\overline{LN}$ intersects $\overline{AD}$ and $\overline{BC}$ at $Z,T$, respectively. Prove that the circumcircle of triangle $\triangle XFY$ and the circle with diameter $EI$ are tangent if and only if the circumcircle of triangle $\triangle TEZ$ and the circle with diameter $FI$ are tangent. [i]Proposed by Mahdi Etesamifard[/i]

Let $M(n)$ and $m(n)$ are maximal and minimal proper divisors of $n$ Natural number $n>1000$ is on the board. Every minute we replace our number with $n+M(n)-m(n)$. If we get prime, then process is stopped. Prove that after some moves we will get number, that is not divisible by $17$

In a certain circle, the chord of a $d$-degree arc is 22 centimeters long, and the chord of a $2d$-degree arc is 20 centimeters longer than the chord of a $3d$-degree arc, where $d<120.$ The length of the chord of a $3d$-degree arc is $-m+\sqrt{n}$ centimeters, where $m$ and $n$ are positive integers. Find $m+n.$

Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.

Construct the quadrilateral $ ABCD $ given the lengths of the sides $ AB $ and $ CD $ and the angles of the quadrilateral.

A post service of some country uses carriers to transport the mail, each carrier’s task is to bring the mail from one city to a neighbouring city. It is known that it is possible to send mail from any city to the capital $P$ . For any two cities $A$ and $B$, call $B$ [i]more important than[/i] $A$, if every possible route of mail from $A$ to the capital $P$ goes through $B$. a) Prove that, for any three different cities $A, B$, and $C$, if $B$ is more important than $A$ and $C$ is more important than $B$, then $C$ is more important than $A$. b) Prove that, for any three different cities $A, B$, and $C$, if both B and C are more important than $A$, then either $C$ is more important than $B$ or $B$ is more important than $C$.

Let $f(x)$ and $g(x)$ be two polynomials with real coefficients such that for infinitely many rational values of $x$, the fraction $\frac{f(x)}{g(x)}$ is rational. Prove that $\frac{f(x)}{g(x)}$ can be written as the ratio of two polynomials with rational coefficients.