Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

An engineer is designing an engine. Each cycle, it ignites a negligible amount of fuel, releasing $ 2000 \, \text{J} $ of energy into the cubic decimeter of air, which we assume here is gaseous nitrogen at $ 20^\circ \, \text{C} $ at $ 1 \, \text{atm} $ in the engine in a process which we can regard as instantaneous and isochoric. It then expands adiabatically until its pressure once again reaches $ 1 \, \text{atm} $, and shrinks isobarically until it reaches its initial state. What is the efficiency of this engine? [i]Problem proposed by B. Dejean[/i]

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?

Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be on the lines $AB$ and $AD$, respectively, so that $B$ lies between $A$ and $E$, and $D$ lies between $A$ and $F$. Suppose that $\angle BCE = 20^o$ and $\angle DCF = 25^o$. Find the area of triangle $\vartriangle EAF$.

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. [i]Nikolay Belukhov[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$. [i]Proposed by Nikolay Beluhov[/i] EDIT: It was the problem 3 (not 2), corrected the source title.

There are given $111$ coins and a $n\times n$ table divided into unit cells. This coins are placed inside the unit cells (one unit cell may contain one coin, many coins, or may be empty), such that the difference between the number of coins from two neighbouring cells (that have a common edge) is $1$. Find the maximal $n$ for this to be possible.

Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN, BC$ and line through $H$ and perpendicular to $AN$ intersect each other and form triangle $X Y Z$. Prove that the circumcircle of triangle $X Y Z$ is tangent to $(O)$.

$\overline{5654}_b$ is a power of a prime number. Find $b$ if $b > 6$.

Let $AB$ be a chord of the semicircle $O$ (not the diameter). $M$ is the midpoint of $AB$, and $D$ is a point lies on line $OM$ ($D$ is outside semicircle $O$). Line $l$ passes through $D$ and is parallel to $AB$. $P, Q$ are two points lie on $l$ and $PO$ meets semicircle $O$ at $C$. If $\angle PCD=\angle DMC$, and $M$ is the orthocentre of $\triangle OPQ$. Prove that the intersection of $AQ$ and $PB$ lies on semicircle $O$.

A [i]weird checkerboard[/i] is a coloring of an $8\times8$ grid constructed by making some (possibly none or all) of the following $14$ cuts: [list] [*] the $7$ vertical cuts along a gridline through the entire height of the board, [*] and the $7$ horizontal cuts along a gridline through the entire width of the board. [/list] The divided rectangles are then colored black and white such that the bottom left corner of the grid is black, and no two rectangles adjacent by an edge share a color. Compute the number of weird checkerboards that have an equal amount of area colored black and white. [center] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f768a7a51c9c9bc56a1d55427c33e15e4bcd74.png[/img] [/center]

A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner. [i]Proposed by Nathan Ramesh

Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.

The sum of the reciprocals of the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ is: $ \textbf{(A)}\ \frac {1}{a} \plus{} \frac {1}{b} \qquad \textbf{(B)}\ \minus{} \frac {c}{b} \qquad \textbf{(C)}\ \frac {b}{c} \qquad \textbf{(D)}\ \minus{} \frac {a}{b} \qquad \textbf{(E)}\ \minus{} \frac {b}{c}$

Compute the number of positive integer divisors of $2121$ with a units digit of $1$. [i]2021 CCA Math Bonanza Individual Round #1[/i]

Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space. [b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel. [b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?