There are two piles of stones with $1012$ stones each. Ann and Ben play a game. In every move, a player removes two stones from one of the piles and adds one to the other pile. Ann goes first. The first player to remove the last stone in one of the piles wins the game. Which player has a winning strategy and why?

Jenn draws a scalene triangle, and measures the heights from each of the vertices to its opposite side. She discovers that the three height lengths are all roots of the polynomial $x^3 - 3.9 x^2 + 4.4 x - 1.2.$ What is the length of the inradius of the triangle? $\textbf{(A) }\frac{3}{13}\qquad\textbf{(B) }\frac{3}{11}\qquad\textbf{(C) }\frac{2}{7}\qquad\textbf{(D) }\frac{8}{15}\qquad\textbf{(E) }\frac{9}{14}$

Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.

Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$

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.

We will designate by $Z_{(5)}$ a certain subset of the set $Q$ of the rational numbers . A rational belongs to $Z_{(5)}$ if and only if there exist equal fraction to this rational such that $5$ is not a divisor of its denominator. (For example, the rational number $13/10$ does not belong to $Z_{(5)}$ , since the denominator of all fractions equal to $13/10$ is a multiple of $5$. On the other hand, the rational $75/10$ belongs to $Z_{(5)}$ since that $75/10 = 15/12$). Reasonably answer the following questions: a) What algebraic structure (semigroup, group, etc.) does $Z_{(5)}$ have with respect to the sum? b) And regarding the product? c) Is $Z_{(5)}$ a subring of $Q$? d) Is $Z_{(5)}$ a vector space?

$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column h becomes $a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.

A positive integer $k$ is called [i]powerful [/i] if there are distinct positive integers $p, q, r, s, t$ such that $p^2$, $q^3$, $r^5$, $s^7$, $t^{11}$ all divide k. Find the smallest powerful integer.

Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$. Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.

Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.

Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

On the sides $BC$ and $CD$ of the square $ABCD$ of side $1$, are chosen the points $E$, respectively $F$, so that $<$ $EAB$ $=$ $20$ If $<$ $EAF$ $=$ $45$, calculate the distance from point $A$ to the line $EF$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Two right cones each have base radius 4 and height 3, such that the apex of each cone is the center of the base of the other cone. Find the surface area of the union of the cones.

Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

A billiards table is in the figure of regular hexagon $ABCDEF$. $P$ is the midpoint of $AB$. We shut the ball at $P$, then it touches $Q$ on side $BC$, then it touches side $CD,DE,EF,FA$. Finally, the ball touches side $AB$ again. Let $\theta=\angle BPQ$, find the value range of $\theta$.

Solve the equation in postive integers $$x^2+y^2+1998=1997x-1999y.$$

In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.

Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$. (a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root. (b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.

Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number.

Let $\triangle MAB$ be a triangle with circumcenter $O$. $P$ and $Q$ lie on line $AB$ (both interior or exterior) such that $\angle PMA = \angle BMQ$. Let $D$ be a point on the perpendicular line through $M$ to $AB$. $E$ is the second intersection of the two circles $(DAB)$ and $(DPQ)$. The line $MO$ intersects $AB$ at $J$. Show that the circumcenter of $\triangle EMJ$ lies on line $AB$. [i](Proposed by Tan Rui Xuen)[/i]

Let $a, b, c$ be complex numbers such that $a+b+c = 1$, $a^2+b^2+c^2 = 2$ and $a^3+b^3+c^3 = 3$. Find the value of $a^4 + b^4 + c^4$.

Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?

Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.