For any positive integer $x$, let $f(x)=x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m)))=m^{m^{n+2020}}$. Compute the smallest possible value of $n$. [i]Proposed by Luke Robitaille[/i]

Let $n$ be a positive integer and let $x$ and $y$ be positive divisors of $2n^2-1$. Prove that $x+y$ is not divisible by $2n+1$.

Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

$S_1,S_2,\ldots,S_n$ are subsets of $\{1,2,\ldots,10000\}$ which satisfy that, whenever $|S_i| > |S_j|$, the sum of all elements in $S_i$ is less than the sum of all elements in $S_j$. Let $m$ be the maximum number of distinct values among $|S_1|,\ldots,|S_n|$. Find $\left\lfloor\frac{m}{100}\right\rfloor$.

1) Is it possible to place five circles on the plane in such way that each circle has exactly 5 common points with other circles? 2) Is it possible to place five circles on the plane in such way that each circle has exactly 6 common points with other circles? 3) Is it possible to place five circles on the plane in such way that each circle has exactly 7 common points with other circles?

Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.

Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is: \[ \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.\]

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (0)=2022$ and $f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)$ for all $x, y \in \mathbb{R}$.

Determine all triples $(x,y,p)$ of positive integers such that $p$ is prime, $p=x^2+1$ and $2p^2=y^2+1$.

Alan is assigning values to lattice points on the 3d coordinate plane. First, Alan computes the roots of the cubic $20x^3-22x^2+2x+1$ and finds that they are $\alpha$, $\beta$, and $\gamma$. He finds out that each of these roots satisfy $|\alpha|,|\beta|,|\gamma|\leq 1$ On each point $(x,y,z)$ where $x,y,$ and $z$ are all nonnegative integers, Alan writes down $\alpha^x\beta^y\gamma^z$. What is the value of the sum of all numbers he writes down? [i]Proposed by Alan Abraham[/i]

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$. Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$

Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x\in S$ then $(2x\bmod{16})\in S$.

A young baseball player thinks he has hit a home run and gets excited, but, instead, he has just hit it to an outfielder who is just able to catch the ball, and does so at ground level. The ball was hit at a height of 1.5 meters from the ground at an angle $\phi$ above the horizontal axis. The catch was taken at a horizontal distance 30 meters from home plate, which was where the batter hit the ball. The ball left the bat at a speed of 21 m/s. Find all possible values $0<\phi<90^\circ$, in degrees, rounded to the nearest integer. You may use WolframAlpha, Mathematica, or a graphing aid to compute $\phi$ after you derive an expression to solve for it. [i](Proposed by Ahaan Rungta)[/i]

The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum.

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

Real numbers $x,y,z$ satisfy the inequalities $$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$.

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

Let $n$ be a positive integer. Prove that $$\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n $$ where $(b_1, b_2, ..., b_n)$ is any permutation of the positive real numbers $a_1, a_2, ..., a_n$.

Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge. [img]https://cdn.artofproblemsolving.com/attachments/6/2/50377e7ad5fb1c40e36725e43c7eeb1e3c2849.png[/img] Sophie colors every triangle side in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?

Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$.\\ Prove that $I$, $A$, $M$ are collinear.

The sum of the two $ 5$-digit numbers $ AMC10$ and $ AMC12$ is $ 123422$. What is $ A\plus{}M\plus{}C$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously: $\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$; $\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$; $\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$. (Note that $x_1,x_2,x_3$ may be real or complex numbers.)

Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is [i]merry[/i] if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words \[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\] contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard. [size=6][url]http://www.hpmor.com/chapter/12[/url][/size]