In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$

Let $A'$ be a point on one of the sides of the trapezoid $ABCD$ such that line $AA'$ divides the area of the trapezoid in half. Points $B'$, $C'$, $D'$ are defined similarly. Prove that the intersection points of the diagonals of quadrilaterals $ABCD$ and $A'B'C'D'$ are symmetrical wrt the midpoint of midline of trapezoid $ABCD$.

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares. [asy] unitsize (2 cm); pair A, B, C, D, Bp, Cp, Dp, P; A = (0,0); B = (-1,0); C = (-1,1); D = (0,1); Bp = rotate(-30)*(B); Cp = rotate(-30)*(C); Dp = rotate(-30)*(D); P = extension(C, D, Bp, Cp); fill(A--Bp--P--D--cycle, gray(0.8)); draw(A--B--C--D--cycle); draw(A--Bp--Cp--Dp--cycle); label("$30^\circ$", (-0.5,0.1), fontsize(10)); [/asy]

Find the maximum positive integer $k$ such that there exist $k$ positive integers which do not exceed $2017$ and have the property that every number among them cannot be a power of any of the remaining $k-1$ numbers.

On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible.

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$

Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the midpoints of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

Prove that for any positive real numbers $a, b, c$, $$2(a^3 + b^3 + c^3 + abc) \ge (a+b)(b + c)(c + a)$$.

Let $ABC$ be a triangle. There exists a positive real number $x$ such that $AB=6x^2+1$ and $AC = 2x^2+2x$, and there exist points $W$ and $X$ on segment $AB$ along with points $Y$ and $Z$ on segment $AC$ such that $AW=x$, $WX=x+4$, $AY=x+1$, and $YZ=x$. For any line $\ell$ not intersecting segment $BC$, let $f(\ell)$ be the unique point $P$ on line $\ell$ and on the same side of $BC$ as $A$ such that $\ell$ is tangent to the circumcircle of triangle $PBC$. Suppose lines $f(WY)f(XY)$ and $f(WZ)f(XZ)$ meet at $B$, and that lines $f(WZ)f(WY)$ and $f(XY)f(XZ)$ meet at $C$. Then the product of all possible values for the length of $BC$ can be expressed in the form $a + \dfrac{b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $c$ squarefree and $\gcd (b,d)=1$. Compute $100a+b+c+d$. [i]Proposed by Vincent Huang[/i]

Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ? [i]A. Khrabrov[/i]

Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

A cell phone plan costs $\$20$ each month, plus $5$¢ per text message sent, plus 10¢ for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? $ \textbf{(A)}\$ 24.00\qquad\textbf{(B)}\$ 24.50\qquad\textbf{(C)}\$25.50\qquad\textbf{(D)}\$28.00\qquad\textbf{(E)}\$30.00 $

Each square in a $3\times 3$ grid of squares is colored red, white, blue, or green so that every $2\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\\ [asy] size(8cm); pen grey1, grey2, grey3; grey1 = RGB(211, 211, 211); grey2 = RGB(173, 173, 173); grey3 = RGB(138, 138, 138); for(int i = 0; i < 4; ++i) { draw((i, 0)--(i, 3)); draw((0, i)--(3, i)); } filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1); label('B', (5.5, 2.5)); filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2); label('R', (6.5, 2.5)); filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1); label('B', (7.5, 2.5)); filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3); label('G', (5.5, 1.5)); filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white); filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3); label('G', (7.5, 1.5)); filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2); label('R', (5.5, 0.5)); filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1); label('B', (6.5, 0.5)); filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2); label('R', (7.5, 0.5)); [/asy] $\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96$

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

There is a row of $100$ cells each containing a token. For $1$ dollar it is allowed to interchange two neighbouring tokens. Also it is allowed to interchange with no charge any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the reverse order? (Egor Bakaev)

At the beginning of a game, I write the numbers $1$, $2$, ..., $2004$ onto a desk. A move consists of - selecting some numbers standing on the desk; - calculating the rest of the sum of these numbers under division by $11$ and writing this rest onto the desk; - deleting the selected numbers. In such a game, after a number of moves, only two numbers remained on the desk. One of them was $1000$. What was the other one?

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

We have a $2\times n$ rectangle. We call each $1\times1$ square a room and we show the room in the $i^{th}$ row and $j^{th}$ column as $(i,j)$. There are some coins in some rooms of the rectangle. If there exist more than $1$ coin in each room, we can delete $2$ coins from it and add $1$ coin to its right adjacent room OR we can delete $2$ coins from it and add $1$ coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room $(1,n)$.

Circle I passes through the center of, and is tangent to, circle II. The area of circle I is 4 square inches. Then the area of circle II, in square inches, is: $ \textbf{(A) }8\qquad\textbf{(B) }8\sqrt{2}\qquad\textbf{(C) }8\sqrt{\pi}\qquad\textbf{(D) }16\qquad\textbf{(E) }16\sqrt{2} $

Let $f(n)$ be the sum of the first $n$ terms of the sequence $0,1,1,2,2,3,3,4, \dots,$ where the $n$th term is given by $$a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}$$ Show that if $x$ and $y$ are positive integers and $x>y$ then $xy=f(x+y)-f(x-y)$.

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna? $\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$

For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$