The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?

Find all real numbers $a$ and $b$ that satisfy the system of equations: $$\begin{cases} a &= \frac{2}{a+b} \\ \\ b &= \frac{2}{3a-b} \\ \end{cases}$$

A number is said to be regular if when a digit $k$ appears in that number, the digit appears exactly $k$ times. For example, the number $3133$ is a regular number because the digit $1$ appears exactly once and the digit $3$ appears exactly three times. How many regular six-digit numbers are there?

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

Anna wants to form a four-digit number with four different digits from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$. She wants the first digit of that number to be bigger than the sum of the other three digits. How many such numbers can she form?

Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.

In the cells of a $5 \times 5$ grid there are some lamps. If a lamp is touched, it is turned on and it lights up all of its neighbouring cells, including its own cell. If a cell is lit up and there is a lamp in it, the lamp is also turned on and lights up its neighbouring cells, including its own. What is the smallest number of lamps needed to light up all of the cells with just one touch? (Note: Two cells are neighbours if they have a common side or vertex.)

Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.

On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.

Show that for any real numbers $a$ and $b$ different from $0$, the inequality $$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$holds. When is equality achieved?

Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

Let $h_a$, $h_b$ and $h_c$ be the altitudes of a triangle $\triangle ABC$ ejected from the vertices $A$,$B$ and $C$, respectively. Similarly, let $h_x$, $h_y$ and $h_z$ be the altitudes of an another triangle $\triangle XYZ$. Show that if $$h_a : h_b : h_c = h_x : h_y : h_z, $$then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.

A subset $S$ of the natural numbers is called dense for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

For a sequence of integers $a_1 < a_2 < \cdot\cdot\cdot < a_n$, a pair $(a_i,a_j)$ where $1 \leq i < j \leq n$ is said to be balanced if the number $\frac{a_i+a_j}{2}$ belongs to the sequence. For every natural number $n \geq 3$, find the maximum possible number of balanced pairs in a sequence with $n$ numbers.

An $n \times n$ board is given. In the top left corner cell there is a fox, whereas in the bottom left corner cell there is a rabbit. Every minute, the fox and the rabbit jump to a neighbouring cell at the same time. The fox can jump only to neighbouring cells that are below it or on its right, whereas the rabbit can only jump to the cells above it or in its right. They continue like this until they have no possible moves. The fox catches the rabbit if at a certain moment they are in the same cell, otherwise the rabbit gets away. Find all natural numbers $n$ for which the fox has a winning strategy to catch the rabbit. (Note: Two squares are considered neighbours if they have a common side.)

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that $$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.

We say that a digit is high if it is placed between two other digits and it is bigger than both of them. The digits $0$,$1$,$2$,$\dots$,$9$ are used exactly once to form a 10-digit number. How many numbers can be formed with the property such that they don’t have any high digits?

Find the smallest natural number $k$ such that the system of equations $$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$has only one solution for positive real numbers $x$, $y$ and $z$.

Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively. Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if $$g_a : g_b : g_c = g_x : g_y : g_z, $$then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.