The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$. (Folklore)

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.

On the sides $AB{}$ and $AC{}$ of the acute-angled triangle $ABC{}$ the points $M{}$ and $N{}$ are chosen such that $MN$ passes through the circumcenter of $ABC.$ Let $P{}$ and $Q{}$ be the midpoints of the segments $CM{}$ and $BN{}.$ Prove that $\angle POQ=\angle BAC.$

Compute \[ \left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right). \] [i]2020 CCA Math Bonanza Team Round #4[/i]

Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that \[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\] Prove that \[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\] [i]Cezar Lupu[/i]

Let $ABCD$ be an isosceles trapezoid such that $AB=10$, $BC=15$, $CD=28$, and $DA=15$. There is a point $E$ such that $\triangle AED$ and $\triangle AEB$ have the same area and such that $EC$ is minimal. Find $EC$.

The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B, C, N,$ and $L$ are concyclic.

Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$

Starting from a given cyclic quadrilateral $\mathcal{Q}_0$, a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$, $B$, $C$ and $D$.) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.

On the side $BC$ of the triangle $ABC$ a point $D$ different from $B$ and $C$ is chosen so that the bisectors of the angles $ACB$ and $ADB$ intersect on the side $AB$. Let $D'$ be the symmetrical point to $D$ with respect to the line $AB$. Prove that the points $C, A$ and $D'$ are on the same line.

Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$. [i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]

Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied: (a) $xf(x,y,z) = zf(z,y,x)$, (b) $f(x, ky, k^2z) = kf(x,y,z)$, (c) $f(1, k, k+1) = k+1$. ([i]United Kingdom[/i])

Consider a square painting of size $1 \times 1$. A rectangular sheet of paper of area $2$ is called its “envelope” if one can wrap the painting with it without cutting the paper. (For instance, a $2 \times 1$ rectangle and a square with side $\sqrt2$ are envelopes.) a) Show that there exist other envelopes. (4) b) Show that there exist infinitely many envelopes. (3)

Let $\vartriangle ABC$ be a triangle with integer side lengths such that $BC = 2016$. Let $G$ be the centroid of $\vartriangle ABC$ and $I$ be the incenter of $\vartriangle ABC$. If the area of $\vartriangle BGC$ equals the area of $\vartriangle BIC$, find the largest possible length of $AB$.

Find the greatest prime factor of $2^{56} + (2^{15} + 1)(2^{29} + 2^{15} + 1)$.

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Prove that for each positive integer $n$, there exists a number $M$, such that $M$ can be written as sum of $1,2,3,\dots, n$ distinct perfect squares.

Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$. Prove that $x + y + z$ is an integer.

Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? $ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $

Is there any sequence $(a_n)_{n=1}^{\infty}$ of non-negative numbers, for which $\sum_{n=1}^{\infty} a_n^2<\infty$ , but $\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty$ ? [hide=Remark]That contest - Miklos Schweitzer 2010- is missing on the contest page here for now being. The statements of all problems that year can be found [url=http://www.math.u-szeged.hu/~mmaroti/schweitzer/]here[/url], but unfortunately only in Hungarian. I tried google translate but it was a mess. So, it would be wonderful if someone knows Hungarian and wish to translate it. [/hide]