$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]

For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points. A function $f_k: S \to N$ is called a [i]coloring function[/i] if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$

Find the addition of all positive integers n that follows the following: $ \frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}} $ is an integer.

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. [i]Nanang Susyanto, Jogjakarta [/i]

An equilateral triangle of integer side length $n \geq 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for $n = 5$. In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines. [img]https://cdn.artofproblemsolving.com/attachments/e/9/17e83ad4872fcf9e97f0479104b9569bf75ad0.jpg[/img] It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let $f(n)$ denote the number of distinct colorings satisfying this condition. Determine, with proof, $f(n)$ for every $n \geq 1$

Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.

Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

There are $101$ not necessarily different weights, each of which weighs an integer number of grams from $1$ g to $2020$ g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than $2020$. What is the largest number of grams can weigh all $101$ weights? (Bogdan Rublev)

Prove that two isotomic lines of a triangle cannot meet inside its medial triangle. [i](Two lines are isotomic lines of triangle $ABC$ if their common points with $BC, CA, AB$ are symmetric with respect to the midpoints of the corresponding sides.)[/i]

Three runners start running simultaneously from the same point on a $500$-meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$, $4.8$, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run? $ \textbf{(A)}\ 1,000 \qquad\textbf{(B)}\ 1,250 \qquad\textbf{(C)}\ 2,500 \qquad\textbf{(D)}\ 5,000 \qquad\textbf{(E)}\ 10,000 $

The triangle $ ABC$ with the circumcircle $ k(U,r)$ is given. On the extension of the radii $ UA$ a point $ P$ is chosen. The reflection of the line $ PB$ on the line $ BA$ is called $ g$. Likewise the reflection of the line $ PC$ on the line $ CA$ is called $ h$. The intersection of $ g$ and $ h$ is called $ Q$. Find the geometric location of all possible intersections $ Q$, while $ P$ passes through the extension of the radii $ UA$.

Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.

How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?

Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$ When does equality holds?

Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

Assume every $ 7$-digit whole number is a possible telephone number except those which begin with $ 0$ or $ 1$. What fraction of telephone numbers begin with $ 9$ and end with $ 0$? \[ \textbf{(A)}\ \frac{1}{63} \qquad \textbf{(B)}\ \frac{1}{80} \qquad \textbf{(C)}\ \frac{1}{81} \qquad \textbf{(D)}\ \frac{1}{90} \qquad \textbf{(E)}\ \frac{1}{100} \]

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

Given scales and a set of $n$ different weights. We take weights in turn and add them on one of the scales sides. Let us denote "$L$" the scales state with the left side down, and "$R$" -- with the right side down. a) Prove that you can arrange the weights in such an order, that we shall obtain the sequence $LRLRLRLR...$ of the scales states. (That means that the state of the scales will be changed after putting every new weight.) b) Prove that for every $n$-letter word containing $R$'s and $L$'s only you can arrange the weights in such an order, that the sequence of the scales states will be described by that word.

Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$? [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(8pt)); fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray); xaxis("$x$",-0.5,4,EndArrow(HookHead,4)); yaxis("$y$",-0.5,4,EndArrow(4)); draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0)); draw((1,0)--(1,2)--(3,2)); draw((2/3,0)--(3,3)); label("$(a,0)$",(2/3,0),S); label("$(3,3)$",(3,3),NE);[/asy]$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac35\qquad \textbf{(C)}\ \frac23\qquad \textbf{(D)}\ \frac34\qquad \textbf{(E)}\ \frac45$

2011 storage buildings are connected by roads so that it is possible to reach any building from any other building, possibly using multiple roads. The buildings contain $x_1,\dots,x_{2011}$ kilogram of cement. In one move, it is possible to relocate any quantity of cement from one building to any other building that is connected to it. The target is to have $y_1,\dots,y_{2011}$ redistributed across storage buildings and \[x_1+x_2+\dots+x_{2011}=y_1+y_2+\dots+y_{2011}.\] What is the minimal number of moves that the redistribution can take regardless of values of $x_i$ and $y_i$ and of the road plan? (Author: P. Karasev)