Let $\alpha, \beta$ be real numbers such that $\sin\alpha\sin\beta=\frac{1}{3}$. Prove that the set of possible values of $\cos \alpha \cos \beta$ is the interval $\left[-\frac{2}{3}, \frac{2}{3}\right]$.

Let $a, b, c$ be positive real numbers. Show that $$a^5+b^5+c^5 \geq 5abc(b^2-ac)$$ and determine when the equality occurs.

In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.

Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at a point $E$. Prove that the projections of $E$ on $AB,BC,CD,DA$ are concyclic.

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.

Lines \( FD \) and \( BE \) intersect at point \( O \). Rays \( OA \) and \( OC \) are drawn from point \( O \). You are given the following information about the angles: \[ \angle DOC = 36^\circ, \quad \angle AOC = 90^\circ, \quad \angle AOB = 4x, \quad \angle FOE = 5x, \] as shown in the figure below. What is the degree measure of \( x \)? [img]https://i.ibb.co/m5rwmXm/Kyiv-MO-2025-R1-7.png[/img]

Let $ ABC$ be an equilateral triangle of center $ O$, and $ M\in BC$. Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$.

Solve the system of equations $x+y+z=\pi$ $\tan x\tan z=2$ $\tan y\tan z=18$

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

In $\triangle ABC$ with circumcenter $O$, $\measuredangle A = 45^\circ$. Denote by $X$ the second intersection of $\overrightarrow{AO}$ with the circumcircle of $\triangle BOC$. Compute the area of quadrilateral $ABXC$ if $BX = 8$ and $CX = 15$. [i]Proposed by Aaron Lin[/i]

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.

Let $n \geq 2$ be a positive integer and define $k$ to be the number of primes $\leq n$. Let $A$ be a subset of $S = \{2,...,n\}$ such that $|A| \leq k$ and no two elements in $A$ divide each other. Show that one can find a set $B$ such that $|B| = k$, $A \subseteq B \subseteq S$ and no two elements in $B$ divide each other.

Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

In a row there are $12$ cards that can be of three kinds: with both white faces, with both black faces or with one white face and the other black. Initially there are $9$ cards with the black side facing up. The first six cards from the left are turned over, leaving $9$ cards with the black face up. The six cards on the left are then turned over, leaving $8$ cards with the black face up. Finally, six cards are turned over: the first three on the left and the last three on the right, leaving $3$ cards with the black face up. Decide if with this information it is possible to know with certainty how many cards of each kind are in the row

An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation \[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \] for all $n\ge 1$. Prove that $a_1 < 2^m$.

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$.

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous. [b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.

In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.