How many ways are there to arrange three indistinguishable rooks on a $ 6 \times 6$ board such that no two rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)

A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

Let $N$ be the number of non-empty subsets $T$ of $S = \{1,2, 3,4,...,2020\}$ satisfying $max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

A set $S$ of positive integers is said to be [i]channeler[/i] if for any three distinct numbers $a,b,c \in S$, we have $a\mid bc$, $b\mid ca$, $c\mid ab$. a) Prove that for any finite set of positive integers $ \{ c_1, c_2, \ldots, c_n \} $ there exist infinitely many positive integers $k$, such that the set $ \{ kc_1, kc_2, \ldots, kc_n \} $ is a channeler set. b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.

A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases: (a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.) (b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen). Remark: If $A$ consists of just one square, then $k = 1$ and the required way is the following: all kings to the left of the square of $A$ make one move to the right.

Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$ \left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Let $\ a,x,y,z>0$. Prove that: $\frac{a+y}{a+z}x+\frac{a+z}{a+x}y+\frac{a+x}{a+y}z\geq{x+y+z}\geq\frac{a+z}{a+x}x+\frac{a+x}{a+y}y+\frac{a+y}{a+z}z$

Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$. Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]

For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.

Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly one of the trinomials $x^2-ax+b$ or $x^2-bx+a$ from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.

The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?

Let $ABC$ be a triangle inscribed in $(O)$. Two tangents of $(O)$ at $B,C$ meets at $P$. The bisector of angle $BAC $ intersects $(P,PB)$ at point $E$ lying inside triangle $ABC$. Let $M,N$ be the midpoints of arcs $BC$ and $BAC$. Circle with diameter $BC$ intersects line segment $EN$ at $F$. Prove that the orthocenter of triangle $EFM$ lies on $BC$.

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

$\triangle ABC$ is a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ is the bisector of $\angle ABC$, then $\angle BDC =$ [asy] size(200); defaultpen(linewidth(0.8)+fontsize(11pt)); pair A= origin, B = 3 * dir(25), C = (B.x,0); pair X = bisectorpoint(A,B,C), D = extension(B,X,A,C); draw(B--A--C--B--D^^rightanglemark(A,C,B,4)); path g = anglemark(A,B,D,14); path h = anglemark(D,B,C,14); draw(g); draw(h); add(pathticks(g,1,0.11,6,6)); add(pathticks(h,1,0.11,6,6)); label("$A$",A,W); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,S); label("$20^\circ$",A,8*dir(12.5)); [/asy] $ \textbf{(A)}\ 40^\circ \qquad \textbf{(B)}\ 45^\circ \qquad \textbf{(C)}\ 50^\circ \qquad \textbf{(D)}\ 55^\circ \qquad \textbf{(E)}\ 60^\circ $

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

Let $ABC$ be a triangle with $AB=AC$. Let point $Q$ be on plane such that $AQ \parallel BC$ and $AQ = AB$. Now let the $P$ be the foot of perpendicular from $Q$ to $BC$. Show that the circle with diameter $PQ$ is tangent to the circumcircle of triangle $ABC$.

Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.

Find all possible triples of integers $a, b, c$ satisfying $a+b-c = 1$ and $a^2+b^2-c^2 =-1$.

You have an unlimited supply of square tiles with side length $ 1$ and equilateral triangle tiles with side length $ 1$. For which n can you use these tiles to create a convex $n$-sided polygon? The tiles must fit together without gaps and may not overlap.

Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ\equal{} 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$.

The points $P$ and $Q$ on the side $AC$ of the non-isosceles $\Delta ABC$ are such that $\angle ABP=\angle QBC<\frac{1}{2}\angle ABC$. The angle bisectors of $\angle A$ and $\angle C$ intersect the segment $BP$ in points $K$ and $L$ and the segment $BQ$ in points $M$ and $N$, respectively. Prove that $AC$,$KN$, and $LM$ are concurrent.