$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

A ladder is a non-decreasing sequence $a_1, a_2, \dots, a_{2020}$ of non-negative integers. Diego and Pablo play by turns with the ladder $1, 2, \dots, 2020$, starting with Diego. In each turn, the player replaces an entry $a_i$ by $a_i'<a_i$, with the condition that the sequence remains a ladder. The player who gets $(0, 0, \dots, 0)$ wins. Who has a winning strategy? [i]Proposed by Violeta Hernández[/i]

Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.

For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements. ($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$. ($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$. ($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$. $ (a)$ Show that if all six of these statements are true then the hexagon is cyclic. $ (b)$ Prove that, in fact, five of the six statements suffice.

Let $X$ be a set with $n$ elements and $0 \le k \le n$. Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$). Let $b_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at most $k$ common components. (a) Show that $a_{n,k} \cdot b_{n,k-1} \le n!$. (b) Let $p$ be prime, and find the exact value of $a_{p,2}$.

The numbers $1,2,\dots,4022$ are placed to the cells of a $2 \times 2011$ chessboard in such a way that successive numbers should be inside cells with common sides. How many such arrangements are there? $\textbf{(A)}\ 16168444 \qquad\textbf{(B)}\ 12168440 \qquad\textbf{(C)}\ 10088242 \qquad\textbf{(D)}\ 8084224 \qquad\textbf{(E)}\ \text{None}$

Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave." [i]L. Fejes- Toth, E. Makai[/i]

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$, prove that $BE = CF$. (Oleksii Karliuchenko)

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $

Prove that $1^{2011}+2^{2011}+3^{2011}+...+2012^{2011} $ is divisible by $2025078$.

The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.

Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest? $\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 960 \qquad \textbf{(C)}\ 1120 \qquad \textbf{(D)}\ 1920 \qquad \textbf{(E)}\ 3840$

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

On each of the $2014^2$ squares of a $2014 \times 2014$-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least $1007$ light bulbs are on and changing the state of all $2014$ light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer $k$ such that from each starting situation there is a finite sequence of moves to a situation in which at most $k$ light bulbs are on.

General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!). To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns. Both of them must give a command and execute it in their turn. Only two commands are possible: The command "[i]Divide![/i]" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier. On the other hand, the command "[i]Rule![/i]" removes exactly one tin soldier from each company. The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command. a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each? b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?

Sam spends his days walking around the following $2\times 2$ grid of squares. \begin{tabular}[t]{|c|c|}\hline 1&2\\ \hline 4&3 \\ \hline \end{tabular} Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?

Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.

Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$. Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.

Let $f(n)$ denote the number of permutations $(a_{1}, a_{2}, \ldots ,a_{n})$ of the set $\{1,2,\ldots,n\}$, which satisfy the conditions: $a_{1}=1$ and $|a_{i}-a_{i+1}|\leq2$, for any $i=1,2,\ldots,n-1$. Prove that $f(2006)$ is divisible by 3.

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

The vertices of a cube have coordinates $(0,0,0),(0,0,4),(0,4,0),(0,4,4),(4,0,0)$,$(4,0,4),(4,4,0)$, and $(4,4,4)$. A plane cuts the edges of this cube at the points $(0,2,0),(1,0,0),(1,4,4)$, and two other points. Find the coordinates of the other two points.

A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure. a) Show that the triangles $AOB$ and $COD$ have the equal areas. b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.

The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.