Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.

Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers. (a) Prove that the expression $$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients. (b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.

Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA = 90^{\circ}$. Given that $AI = 97$ and $BC = 144$, compute the area of $\triangle ABC$.

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

Let $a$ be a positive real number and let $b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$. Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

Let $M$ and $N$ be two points on different sides of the square $ABCD$. Suppose that segment $MN$ divides the square into two tangential polygons. If $R$ and $r$ are radii of the circles inscribed in these polygons ($R> r$), calculate the length of the segment $MN$ in terms of $R$ and $r$. (Moldova)

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label("$A$",A,W); label("$B$",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi\minus{}\sqrt3 \qquad \textbf{(B)}\ \pi\minus{}\sqrt2 \qquad \textbf{(C)}\ \frac{\pi\plus{}\sqrt2}{2} \qquad \textbf{(D)}\ \frac{\pi\plus{}\sqrt3}{2}$ $ \textbf{(E)}\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}$

Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$ [i]Proposed by Joshua Hsieh[/i]

Find the ordered triple of natural numbers $(x,y,z)$ such that $x \le y \le z$ and $x^x+y^y+z^z = 3382.$ [i]Proposed by Evan Fang

Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$

For a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers that are at least $ 1, $ prove that the series $ \sum_{i=1}^{\infty } \frac{1}{x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{1+x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{\lfloor x_i\rfloor } $ converges.

For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions. [list] [*] $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$ [*] $x_1+x_2+ \cdots +x_k = n$ [*] $x_1-x_k \le r$ [/list] For all positive integers $m, s, t$, prove that $$A(m, s, t)=A(m, t, s).$$

Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$. Prove that $AT=SB$.