$99$ different points $P_1, P_2, ..., P_{99}$ are marked on circle $O$. For each $P_i$, define $n_i$ as the number of marked points you encounter starting from $P_i$ to its antipode, moving clockwise. Prove the following inequality. $$n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900$$

Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number. [b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]

$X$ is a set of $2020$ distinct real numbers. Prove that there exist $a,b\in \mathbb{R}$ and $A\subset X$ such that $$\sum_{x\in A}(x-a)^2 +\sum_{x\in X\backslash A}(x-b)^2\le \frac{1009}{1010}\sum_{x\in X}x^2$$

Two sets of positive integers $A, B$ are called [i]connected [/i] if they are not empty and for all $a \in A, b \in B$, number $ab + 1$ is a perfect square. i) Given $A =\{1, 2,3, 4\}$. Prove that there does not exist any set $B$ such that $A, B$ are connected. ii) Suppose that $A, B$ are connected with $|A|,|B| \ge 2$. For any $a_1 > a_2 \in A$ and $b_1 > b_2 \in B$, prove that $a_1b_1 > 13a_2b_2$.

In a quadrilateral $ABCD$, a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$, respectively. If $(AM) (CP) = (BN) (DQ)$, prove that $ABCD$ is an cyclic quadrilateral.

Let $x\in\mathbb{R}$. What is the maximum value of the following expression: $\sqrt{x-2018} + \sqrt{2020-x}$ ?

Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) \equal{} b, P(b) \equal{} c,$ and $ P(c) \equal{} a$.

Let $ABCDEF$ be a convex hexagon with area $S$ such that $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$ holds, and whose all angles are obtuse and opposite sides are not the same length. Prove that the following inequality holds: $$A_{ABC} + A_{BCD} + A_{CDE} + A_{DEF} + A_{EFA} + A_{FAB} < S$$ , where $A_{XYZ}$ is the area of triangle $XYZ$

\[\int_{\pi/4}^{\pi/2} \tan^{-1}\left(\tan^2(x)\right)\sin(2x)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Sheldon and Bella play a game on an infinite grid of cells. On each of his turns, Sheldon puts one of the following tetrominoes (reflections and rotations aren't permitted) [asy] size(200); draw((0, 0)--(1, 0)--(1, 2)--(0, 2)--cycle); draw((1, 1)--(2, 1)--(2, 3)--(1, 3)--cycle); draw((0,1)--(1,1)); draw((1,2)--(2,2)); draw((5, 0.5)--(6, 0.5)--(6, 1.5)--(5, 1.5)--cycle); draw((6, 0.5)--(7, 0.5)--(7, 1.5)--(6, 1.5)--cycle); draw((6, 1.5)--(7, 1.5)--(7, 2.5)--(6, 2.5)--cycle); draw((7, 1.5)--(8, 1.5)--(8, 2.5)--(7, 2.5)--cycle); [/asy] somewhere on the grid without overlap. Then, Bella colors that tetromino such that it has a different color from any other tetromino that shares a side with it. After $2631$ such moves by each player, the game ends, and Sheldon's score is the number of colors used by Bella. What's the maximum $N$ such that Sheldon can guarantee that his score will be at least $N$?

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to $ \textbf{(A)} \ 50 \qquad \textbf{(B)} \ 52 \qquad \textbf{(C)} \ 54 \qquad \textbf{(D)} \ 56 \qquad \textbf{(E)} \ 58 \qquad$ [asy]unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } }[/asy]

Let $ \{a_k\}$ be a sequence of integers such that $ a_1 \equal{} 1$ and $ a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn$, for all positive integers $ m$ and $ n$. Then $ a_{12}$ is $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 67 \qquad \textbf{(D)}\ 78 \qquad \textbf{(E)}\ 89$

Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$

The cells of a table [b]m x n[/b], $m \geq 5$, $n \geq 5$ are colored in 3 colors where: (i) Each cell has an equal number of adjacent (by side) cells from the other two colors; (ii) Each of the cells in the 4 corners of the table doesn’t have an adjacent cell in the same color. Find all possible values for $m$ and $n$.

Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers.

In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$

A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that $$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$ for all $(x,y)\in \mathbb{R}^{2}.$

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles. [asy] unitsize(1.5 cm); pair A, B, C, D, G, H; A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B; draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, E); dot("$D$", D, NW); dot("$G$", G, NE); dot("$H$", H, SE); [/asy]

In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$. In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$. Prove that $AB=BC+CD$.

For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$ $\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$

A function $D(n)$ of the positive integral variable $n$ is defined by the following properties: $D(1) = 0, D(p) = 1$ if $p$ is a prime, $D(uv) = u D(v) + v D(u)$ for any two positive integers $u$ and $v.$ Answer all three parts below. (i) Show that these properties are compatible and determine uniquely $D(n).$ (Derive a formula for $D(n) /n,$ assuming that $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ where $p_1, p_2, \ldots, p_k$ are different primes.) (ii) For what values of $n$ is $D(n) = n?$ (iii) Define $D^2 (n) = D[D(n)],$ etc., and find the limit of $D^m (63)$ as $m$ tends to $\infty.$

For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

[b]Q8.[/b] Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$. Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$