Olympiad problems

2022 All-Russian Olympiad, 3

Tags: geometry , geometry solved , Russia , All Russian Olympiad , fixed circle , perpendicular lines , Russian

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$. Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$, and the point $Q$ is on the smaller arc $AC$. Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$, from point $Q$ to the line $AC$ and from point $A$ to line $PQ$. Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

2003 All-Russian Olympiad, 2

Tags: geometry , circumcircle , cyclic quadrilateral , geometric transformation , geometry proposed , Spiral Similarity , Russian

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

2003 All-Russian Olympiad, 2

Tags: geometry , circumcircle , cyclic quadrilateral , geometric transformation , geometry proposed , Spiral Similarity , Russian

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

2022 All-Russian Olympiad, 1

Tags: number theory , All russia mo , All Russian Olympiad , Russia , Russian , number theory solved , Divisors

We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

2021 Romanian Master of Mathematics Shortlist, G3

Tags: RMM Shortlist , geometry , Russian

Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle BAC > 90^{\circ}$ and $AB > AC$. The tangents of $\Omega$ at $B$ and $C$ cross at $D$ and the tangent of $\Omega$ at $A$ crosses the line $BC$ at $E$. The line through $D$, parallel to $AE$, crosses the line $BC$ at $F$. The circle with diameter $EF$ meets the line $AB$ at $P$ and $Q$ and the line $AC$ at $X$ and $Y$. Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.