For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

We color numbers $1,2,3,...,20$ in two colors, blue and yellow, such that both colors are used (not all numbers are colored in one color). Determine number of ways we can color those numbers, such that product of all blue numbers and product of all yellow numbers have greatest common divisor $1$.

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

There is one nonzero digit in every cell of $2017\times 2017 $ table. On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$. [hide=Example]Example for $2\times2$. If table is |1|4| |3|7|. Then numbers on board are $14,37,13,47$[/hide]

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$.

A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. (a) If $\theta = 90^o$, determine all locations where the point could end up. (b) If $\theta = 45^o$, prove that for every location $ L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (c) Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (d) Prove that when $\theta$ is irrational, for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L.$

There are $30$ teams in NBA and every team play $82$ games in the year. Bosses of NBA want to divide all teams on Western and Eastern Conferences (not necessary equally), such that number of games between teams from different conferences is half of number of all games. Can they do it?

Balls of $3$ colours — red, blue and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy] $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~\displaystyle\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\displaystyle\frac{4}{5}\qquad\textbf{(D)}~\displaystyle\frac{11}{13}\qquad\textbf{(E)}~\displaystyle\frac{7\sqrt{3}}{13}$

Let $ABCD$ and $AEFG$ be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point $P$. Prove that $P$ lies on line $BE$.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is: $ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $

Given two positive integers $a$ and $b$, an [i]legal move[/i] consists in choosing a proper divisor of one of them and adding it to $a$ or adding it to $b$. Two players, Agustin and Ian, take turns making an legal move; Agustin plays first. Whoever gets a number greater than or equal to $2015$ wins the game. [list=a] [*]Determine which of the players has a winning strategy if $a=3, b=5$. [*]Determine which of the players has a winning strategy if $a=6, b=7$. [/list]

Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

Let $\ell$ be a line and $P$ be a point in $\mathbb{R}^3$. Let $S$ be the set of points $X$ such that the distance from $X$ to $\ell$ is greater than or equal to two times the distance from $X$ to $P$. If the distance from $P$ to $\ell$ is $d>0$, find $\text{Volume}(S)$.

In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)

Let $ABCDEF$ be a regular hexagon of side lengths $1$ and $O$ its centre, Join $O$ cto each of the six vertices , thus getting $12$ unit line segments in all. Find the number of closed paths from (i) $O$ to $O$ (ii) $A$ to $A$ each of length $2004$

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P .

Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients \[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\] form an increasing arithmetic series.

Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.

There $100$ people seated at a round table $50$ women and $50$ men. Show that there are two people of opposite gender that stay between two people of opposite gender. (WWMM, MMWW, WMWM, MWMW)

An equilateral triangle with integer side length $n$ is subdivided into smaller equilateral triangles of side length $1$ by drawing lines parallel to its sides, as shown in the figure for $n = 4$. [asy] size(5cm); // Function to draw an equilateral triangle with subdivisions and mark vertices void drawTriangleWithDots(pair A, pair B, pair C, int n) { real step = 1.0 / n; // Draw horizontal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (C - A); pair end = start + i * step * (B - C); draw(start -- end, gray(0.5)); } // Draw left-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (B - A); pair end = start + (n - i) * step * (C - A); draw(start -- end, gray(0.5)); } // Draw right-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = B + i * step * (C - B); pair end = start + (n - i) * step * (A - B); draw(start -- end, gray(0.5)); } // Mark dots at all vertices for (int i = 0; i <= n; ++i) { for (int j = 0; j <= i; ++j) { pair vertex = A + i * step * (C - A) + j * step * (B - C); dot(vertex, black); } } // Draw the outer triangle draw(A -- B -- C -- cycle, black+linewidth(1)); } // Main triangle vertices pair A = (0, 0); pair B = (4, 0); pair C = (2, 3.464); // Height = sqrt(3)/2 * side length // Subdivisions int n = 4; // Draw the subdivided equilateral triangle with dots drawTriangleWithDots(A, B, C, n); [/asy] Consider the set $A$ consisting of all points that are vertices of any of these smaller triangles. A [i]subtriangle[/i] is defined as any equilateral triangle whose three vertices belong to the set $A$ and whose three sides lie along the lines of the initial subdivision. We wish to color all points in $A$ either red or blue such that no subtriangle has all three vertices of the same color. Let $C(n)$ denote the number of such valid colorings for each positive integer $n$. Calculate, in terms of $n$, the value of $C(n)$.