Let $ n$ be a positive integer, and let $ M \equal{} \{1,2,\ldots, 2n\}$. Find the minimal positive integer $ m$, such that no matter how we choose the subsets $ A_i \subset M$, $ 1\leq i\leq m$, with the properties: (1) $ |A_i\minus{}A_j|\geq 1$, for all $ i\neq j$, (2) $ \bigcup_{i\equal{}1}^m A_i \equal{} M$, we can always find two subsets $ A_k$ and $ A_l$ such that $ A_k \cup A_l \equal{} M$ (here $ |X|$ represents the number of elements in the set $ X$.)

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which \[f(x + y) = f(x - y) + f(f(1 - xy))\] holds for all real numbers $x$ and $y$.

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Determine all positive real solutions $(a,b)$ to the following system of equations. \begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}

The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.

Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have? [img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

Define a [i]rooted tree[/i] to be a tree $T$ with a singular node designated as the [i]root[/i] of $T$. (Note that every node in the tree can have an arbitrary number of children.) Each vertex adjacent to the root node of $T$ is itself the root of some tree called a [i]maximal subtree[/i] of $T$. Say two rooted trees $T_1$ and $T_2$ are [i]similar[/i] if there exists some way to cycle the maximal subtrees of $T_1$ to get $T_2$. For example, the first pair of trees below are similar but the second pair are not. How many rooted trees with $2019$ nodes are there up to similarity? [center] [img=500x100]https://i.imgur.com/8axcDvz.png[/img] [/center]

Let $s$ and $t$ be the solutions to $x^2-10x+10=0$. Compute $\tfrac{1}{s^5}+\tfrac{1}{t^5}$.

A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.

A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have $\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$ Now what?

Consider an equilateral triangle $\vartriangle ABC$ with side length $1$. Let $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $\angle ADE = 30^o$ and $DE$ is tangent to the incircle of $\vartriangle ABC$. Compute the perimeter of $\vartriangle ADE$.

How many integers $n$, satisfy $|n|<2020$ and the equation $11^3|n^3+3n^2-107n+1$ (A)$0$ (B)$101$ (C)$367$ (D)$368$

Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$

Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

When Troy writes his digits, his $0$, $1$, and $8$ look the same right-side-up and upside-down as seen in the figure below. His $6$ and $9$ look like upside-down images of each other. None of his other digits look like digits when they are inverted. How many different five-digit numbers (which do not begin with the digit zero) can Troy write which read the same right-side-up and upside-down? [asy] frame l; label(l,"\textsf{0}\qquad \textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}"); add(rotate(180)*l); label("\textsf{0}\qquad\textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}",(0,20)); [/asy]

Let $C_1$ and $C_2$ be circles of radius $1$ that are in the same plane and tangent to each other. How many circles of radius $3$ are in this plane and tangent to both $C_1$ and $C_2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

For what minimal natural $a$ the polynomial $ax^2 + bx + c$ with the integer $c$ and $b$ has two different positive roots both less than one.

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

$\textbf{Problem C.1}$ There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ($A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?

Find all triples of integers $(x, y, z)$ not zero and relative primes in pairs such that $\frac{(y+z-x)^2}{4x}$, $\frac{(z+x-y)^2}{4y}$ and $\frac{(x+y-z)^2}{4z}$ are all integers.