For a positive integer $n$, an $n$-configuration is a family of sets $\langle A_{i,j} \rangle_{1\le i,j\le n}$. An $n$-configuration is called *sweet* if for every pair of indices $(i, j)$ with $1 \le i \le n-1$ and $1 \le j \le n$ we have $A_{i,j} \subseteq A_{i+1,j}$ and $A_{j,i} \subseteq A_{j,i+1}$. Let $f(n,k)$ denote the number of sweet $n$-configurations such that $A_{n,n} \subseteq \{1, 2, \dots, k\}$. Determine which number is larger: $f(2024, 2024^2)$ or $f(2024^2, 2024)$.
Let $ABC$ be a triangle and $D$ a point on its side $BC$. Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C$, respectively, such that $BE = BD$ and $CF = CD$. Let $P$ be a point such that $D$ is the incenter of triangle $PEF$. Prove that $P$ lies inside the circumcircle $Ω$ of triangle $ABC$ or on it. (Josef Tkadlec, Czech Republic)
Let $ABCD$ be a quadrilateral, such that $AB = BC = CD$. There are points $X, Y$ on rays $CA, BD$, respectively, such that $BX = CY$. Let $P, Q, R, S$ be the midpoints of segments $BX, CY, XD, YA$, respectively. Prove that points $P, Q, R, S$ lie on a circle.
Let $\alpha \neq 0$ be a real number. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(x^2 + y^2) = f(x - y)f(x + y) + \alpha y f(y)$$
holds for all $x, y \in \mathbb{R}$.
Acute triangle $ABC$ with $AB \lt AC$ has circumcircle $\omega$. Let $D$, $E$, $F$ be the midpoints of sides $BC$, $CA$, $AB$ correspondingly. Ray $AD$ intersects $\omega$ at point $K$ ($K \neq A$) and the circumcircle of triangle $AEF$ at point $M$ ($M \neq A$). On circle $\omega$ a point $T$ such that $AT \parallel BC$ is chosen. The circumcircles of triangles $KDT$ and $AEF$ intersect at point $L$ on the shorter arc $EF$. Prove that circle $\omega$ and the circumcircle of triangle $KLM$ are tangent to each other.
Let $n$ and $k$ be positive integers. On the board $k$ different positive integers not exceeding $n$ are written. Prove that the equation
$$
x + y = z + w
$$
has at least $\frac{k^4}{2n-1}$ solutions $(x, y, z, w)$, where $x, y, z, w$ are some (not necessarily different) numbers written on the board.
The number $\sqrt{2}$ is on the computer screen. In one step, Juku can multiply the number currently on the screen by any positive rational number or add any natural number to it. Is there a positive integer $n$ such that every number that Juku can get after a finite number of steps can be obtained by at most $n$ steps? If so, find the least suitable $n$.
Fix a natural number $n$. A function $f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\}$ is called *regular* if $f(0) = 0$ and $f(i) \in \{i-1, f(i-1), f(f(i-1)), \dots\}$ for every $i = 1, \dots, n$. If, for instance, $n = 3$, then the function $f(0) = f(1) = 0, f(2) = f(3) = 1$ is regular, but the function $f(0) = f(1) = f(2) = 0, f(3) = 1$ is not (in the latter case, $f(3)$ violates the regularity condition). Juku chooses a regular function $f$ and tells Miku for each number $k = 0, 1, \dots, n$, how many different arguments $i$ are there such that $f(i) = k$. Can Miku always determine based on this information which regular function Juku had chosen?
A rectangular grid is divided by two perpendicular straight lines into four smaller rectangles with integral side lengths. It is possible to remove one among these four rectangles in such a way that the remaining figure can be exactly covered by rectangles of size $2 \times 3$ and $3 \times 2$. Prove that it is possible to exactly cover one among these four smaller rectangles by rectangles of size $2 \times 3$ and $3 \times 2$. (By *exact covering* we mean covering without gaps, overlaps and overflows.)
In an unopened pack, there are red, green and blue candies that weigh $2$ g, $5$ g and $25$ g respectively. There are the same number of candies of each color. After Mary eats some candies from that pack, the remaining candies weigh exactly $787$ g in total. Find the least number of candies that Mary could have eaten.
In Kükametsa School of Science, the number of secondary school participants in the "Kangaroo" competition was exactly one fourth greater than in the previous year. This year, the number of participants from the 11th and 12th grade was respectively 3 and 4 less than the number of participants from the 10th and 11th grade in the previous year, but the number of participants from the 10th grade this year was twice as high as the number of participants from the 12th grade last year. In the previous year, the average number of participants per grade level in Kükametsa School of Science was more than 60, but less than 65. Find all possibilities how many participants from the 10th grade could have participated in the "Kangaroo" competition this year.
All integers from $1$ to $12$ are written on the edges of a cube so that every edge has exactly one integer. Two robot ants stand in the same vertex of the cube and wish to arrive at the vertex that is furthest away from them. Each ant picks a path consisting of exactly three edges of the cube. They multiply the integers on their path. The product of the integers on the path of the first ant is divisible by $100$, but not by $200$. The sum of the digits of the product of the integers on the path of the second ant is $2$ and the integer $1$ is not on its path. Find all possible products that either ant can get.
a. Does there exist a positive integer $n$ such that the eight last digits of the number $n^2 + 1$ are the same as in the number $2n$, but the ninth digit from the end of these two numbers are different?
b. Does there exist a positive integer $n$ such that the nine last digits of the number $n^2 + 1$ are the same as in the number $2n$, but the tenth digit from the end of these two numbers are different?
On the base $BC$ of the isosceles triangle $ABC$ points $D$ and $E$ are chosen such that $BD = EC = 7$ cm. A line perpendicular to side $AB$ and passing through point $A$ intersects base $BC$ at point $F$. An altitude of triangle $ACE$ is $EG$, a median of triangle $AEF$ is $AH$. It is known that $\angle DAF = \frac{1}{2}\angle BAC$. Find the length of the segment $GH$.
Juku and Miku play the following game on a grid of dimensions $n \times m$: In the beginning, all unit squares are white. Each player on their turn paints one white unit square either red or blue of their choice, but no two unit squares with a common side or a common vertex can be painted the same color. The players take turns and Juku starts. A player who cannot make the allowed move has lost. Is it possible for Juku to win the game regardless of how Miku plays if:
Find the smallest positive integer $n$ for which there exist two distinct pairs of positive integers $(x, y)$ such that $(x^2 - 1)(y^2 - 1) = n$ and $x \le y$.
An acute-angled triangle $ABC$ with $AC \gt AB$ is given. The perpendicular bisector of side $BC$ intersects the lines $AC$ and $AB$ at points $D$ and $E$, respectively. The circle with diameter $DE$ intersects the lines $AC$ and $AB$ at points $K$ and $L$, respectively ($K \neq D, L \neq E$). Let $M$ be the midpoint of side $BC$. Prove that the points $K, L$, and $M$ are collinear.