An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.

Let $ABC$ be a triangle and let $P$ be an interior point of $ABC$. We assume that a line $l$, which passes through $P$, but not through $A$, intersects $AB$ and $AC$ (or their extensions over $B$ or $C$) at $Q$ and $R$, respectively. Find $l$ such that the perimeter of the triangle $AQR$ is as small as possible.

$100$ stones, each weighs $1$ kg or $10$ kgs or $50$ kgs, weighs $500$ kgs in total. How many values can the number of stones weighing $10$ kgs take? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.

How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.)

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

A finite set $\{a_1, a_2, ... a_k\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named [i]alternating [/i] if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\{1, 2, 3,..., n\}$ is denoted by $A(n)$. Develop a method to determine $A(n)$ for every $n \in N$ and calculate hence $A(33)$.

1. Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$. If there are not $10$ consecutive numbers that sum $0$, find all possible values of the sum of the $2012$ numbers.

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

Let $ABC$ be a triangle. The incircle $\omega$ of $\triangle ABC$, which has radius $3$, is tangent to $\overline{BC}$ at $D$. Suppose the length of the altitude from $A$ to $\overline{BC}$ is $15$ and $BD^2 + CD^2 = 33$. What is $BC$?

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.

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.