Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^\circ$. Let the height from $A$ cut its side $BC$ at $D$. Let $I, I_B, I_C$ be the incenters of triangles $ABC, ABD, ACD$ respectively. Let also $E_B, E_C$ be the excenters of $ABC$ with respect to vertices $B$ and $C$ respectively. If $K$ is the point of intersection of the circumcircles of $E_CIB_I$ and $E_BIC_I$, show that $KI$ passes through the midpoint $M$ of side $BC$.
Notation clarification supported by the original diagram: the source writes $B_I,C_I$ where the defined points are $I_B,I_C$. Thus the two circles are $(E_C,I,I_B)$ and $(E_B,I,I_C)$, and $K$ denotes their common point other than $I$, as shown in that diagram. This makes the line $KI$ well-defined.
Can every positive rational number $q$ be written as
$$
\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},
$$
where $a, b, c, d$ are all positive integers?
Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel either clears every piece of rubbish from a single pile, or one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and adds one piece of rubbish to each non-empty pile, or creates a new pile with one piece. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?
Let $ABC$ be an acute triangle with $AC \gt AB$ and circumcircle $\Gamma$. The tangent from $A$ to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$. Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such that $MP$ is parallel to $AB$.
Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$.
There is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual), so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured membrane so that no two friends have the same colour, then there is a way to do it with $2021$ colours, but not with $2020$ or less.
Two friends $A$ and $B$ can decide to *merge* in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign $2020$ colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least $2021$ friends?
Let $ABC$ be an acute triangle such that $AB \lt AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$ and $M$ are collinear.
Let $a$, $b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b] = 2021^c$. If $|a - b|$ is a prime number, prove that the number $(a + b)^2 + 4$ is composite.
Let $\triangle ABC$ be an acute scalene triangle. Its $C$-excircle tangent to the segment $AB$ meets $AB$ at point $M$ and the extension of $BC$ beyond $B$ at point $N$. Analogously, its $B$-excircle tangent to the segment $AC$ meets $AC$ at point $P$ and the extension of $BC$ beyond $C$ at point $Q$. Denote by $A_1$ the intersection point of the lines $MN$ and $PQ$, and let $A_2$ be defined as the point, symmetric to $A$ with respect to $A_1$. Define the points $B_2$ and $C_2$, analogously. Prove that $\triangle ABC$ is similar to $\triangle A_2B_2C_2$.
**Proposed by Bulgaria**
A *super-integer* triangle is defined to be a triangle whose lengths of all sides and at least one height are positive integers. We will deem certain positive integer numbers to be *good* with the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily different) good numbers, then the length of the remaining side is also a good number. Let $5$ be a good number. Prove that all integers larger than $2$ are good numbers.
Let $ABC$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $BC$, $CA$ and $AB$ are $D$, $E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $AI$, and let $L$ be the second point of intersection of the circumcircles of the triangles $BFK$ and $CEK$. If $\frac{1}{3}BC = AC - AB$, prove that $DE = 2KL$.
Let $a$ be a real number. Prove that there exist real numbers $b$ and $c$ such that the inequalities
$$ \min \{\sin(x), \sin(a+x)\} \le b \sin(x+c) \le \max \{\sin(x), \sin(a+x)\} $$
hold for all real numbers $x$ and the equalities hold only if $\sin(x) = \sin(a+x)$.
Given an equilateral triangle, find all positive integers $n$ such that it can be partitioned into $n$ equilateral triangles (not necessarily of the same size).
Let $O$ be the circumcenter of triangle $ABC$. Let $D$, $E$ and $F$ be the reflection of $O$ over lines $BC$, $CA$ and $AB$, respectively. Show that lines $AD$, $BE$ and $CF$ are concurrent.
Let $n$ be a positive integer. Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation
$$ f(x)^n f(x+y) = f(x)^{n+1} + x^n f(y) $$
for all $x, y \in \mathbb{R}$.
Let $t$ be a non-zero real number and $n$ be a positive integer. Let $a_1, a_2, \dots, a_{2n-1}$ be real numbers (not necessarily distinct). Prove that there exist indices $i_1 \lt i_2 \lt \dots \lt i_n$ such that $a_{i_k} - a_{i_{l}} \neq t$ for any $1 \le k, l \le n$.
Point $P$ lies inside triangle $ABC$, $M$ is the midpoint of side $BC$ and $P'$ is symmetric to $P$ with respect to $M$. $K$ and $H$ are projections of $P$ on $AB$ and $AC$ respectively and $KM = HM$. Prove that $\angle PAB = \angle CAP'$.
Let points $A$ and $B$ lie on circle $\omega$ with center $O$. Assume that $O$ does not lie on line $AB$. Let point $C$ lie on segment $AB$ and denote by $M$ and $N$ the midpoints of segments $AC$ and $CB$, respectively. The circumcircle of $AON$ intersects $\omega$ at $A$ and $K$ and the circumcircle of $BOM$ intersects $\omega$ at $B$ and $L$. Moreover, circumcircles of $AON$ and $BOM$ intersect each other at $O$ and $X$. Prove that the quadrilateral $CKXL$ is cyclic.