Let $O$ be the center of a circle circumscribed about an acute angle triangle $ABC$, $S_A$, $S_B$, $S_C$ - circles with center O, tangent to sides $BC$, $CA$, $AB$ respectively. Prove that the sum of three angles : between the tangents to $S_A$ drawn from point $A$, to $S_B$ from point $B$ and to $S_C$ - from point $C$, is equal to $180^o$.

Among $33$ balls, there are $2$ radioactive ones. You can put several balls in the detector and it will show if the both radioactive balls are among the balls. What is the smallest number that we have to use the detector so that one can certainly find at least one of the radioactive balls?

The first $9$ positive integers are placed into the squares of a $3\times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that $a=b=4$?

Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be a point in the plane such that $AP \perp BC$. Let $Q$ and $R$ be the reflections of $P$ in the lines $CA$ and $AB$, respectively. Let $Y$ be the orthogonal projection of $R$ onto $CA$. Let $Z$ be the orthogonal projection of $Q$ onto $AB$. Assume that $H \neq O$ and $Y \neq Z$. Prove that $YZ \perp HO$. [asy] import olympiad; unitsize(30); pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2; A = (-14.8, -6.6); B = (-10.9, 0.3); C = (-3.1, -7.1); O = circumcenter(A,B,C); H = orthocenter(A,B,C); P = 1.2 * H - 0.2 * A; Q = reflect(A, C) * P; R = reflect(A, B) * P; Y = foot(R, C, A); Z = foot(Q, A, B); P2 = foot(A, B, C); Q2 = foot(P, C, A); R2 = foot(P, A, B); draw(B--(1.6*A-0.6*B)); draw(B--C--A); draw(P--R, blue); draw(R--Y, red); draw(P--Q, blue); draw(Q--Z, red); draw(A--P2, blue); draw(O--H, darkgreen+linewidth(1.2)); draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2)); draw(rightanglemark(R,Y,A,10), red); draw(rightanglemark(Q,Z,B,10), red); draw(rightanglemark(C,Q2,P,10), blue); draw(rightanglemark(A,R2,P,10), blue); draw(rightanglemark(B,P2,H,10), blue); label("$\textcolor{blue}{H}$",H,NW); label("$\textcolor{blue}{P}$",P,N); label("$A$",A,W); label("$B$",B,N); label("$C$",C,S); label("$O$",O,S); label("$\textcolor{blue}{Q}$",Q,E); label("$\textcolor{blue}{R}$",R,W); label("$\textcolor{red}{Y}$",Y,S); label("$\textcolor{red}{Z}$",Z,NW); dot(A, filltype=FillDraw(black)); dot(B, filltype=FillDraw(black)); dot(C, filltype=FillDraw(black)); dot(H, filltype=FillDraw(blue)); dot(P, filltype=FillDraw(blue)); dot(Q, filltype=FillDraw(blue)); dot(R, filltype=FillDraw(blue)); dot(Y, filltype=FillDraw(red)); dot(Z, filltype=FillDraw(red)); dot(O, filltype=FillDraw(black)); [/asy]

Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.

In how many ways can we color exactly $k$ vertices of an $n$-gon in red such that any $2$ consecutive vertices are not both red. (Vertices are considered to be labeled)

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.

A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?

A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

Let $n$, $k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there exists a positive integer m which is divisible by $n$ and the sum of its digits in the decimal representation is $k$.

Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be? [i]A. Golovanov[/i]

There are $5$ boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations: i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box, ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box. Is it possible, that after a few steps, obtained conditions where each box contains exactly $17^{5^{2016}}$ balls?