At a party, there are $100$ cats. Each pair of cats flips a coin, and they shake paws if and only if the coin comes up heads. It is known that exactly $4900$ pairs of cats shook paws. After the party, each cat is independently assigned a ``happiness index" uniformly at random in the interval $[0,1]$. We say a cat is [i]practical[/i] if it has a happiness index that is strictly greater than the index of every cat with which it shook paws. The expected value of the number of practical cats is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$. [i]Proposed by Brandon Wang[/i]

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Show that it is possible to arrange seven distinct points in the plane so that among any three of these seven points, two of the points are a unit distance apart. (Your solution should include a carefully prepared sketch of the seven points, along with all segments that are of unit length.)

We consider the sum of the digits of a positive integer. For example, the sum of the digits of $2007$ is equal to $9$, since $2 + 0 + 0 + 7 = 9$. (a) How many integers $n$, where $0 < n < 100 000$, have an even sum of digits? (b) How many integers $n$, where $0 < n < 100 000$, have a sum of digits that is less than or equal to $22$?

The sidelengths of a triangle are natural numbers multiples of $7$, smaller than $40$. How many triangles satisfy these conditions?

Solve the equation in positive integers: $2^x-3^y 5^z=1009$.

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.