Metropolises 2016-21 (IOM) (Russia) 10p

geometry problems from International Olympiad of Metropolises (IOM)
with aops links in the names

International Olympiad of Metropolises all 2016-21 in pdf with solutions

collected inside aops here

2016 - 2021

International Olympiad of Metropolises 2016 P3

Let A₁A₂ . . .A_n be a cyclic convex polygon whose circumcenter is strictly in its interior. Let B₁, B₂, …, B_n be arbitrary points on the sides A₁A₂, A₂A₃, …, A_nA₁, respectively, other than the vertices. Prove that $$\frac{{{B}_{1}}{{B}_{2}}}{{{A}_{1}}{{A}_{3}}}+\frac{{{B}_{2}}{{B}_{3}}}{{{A}_{2}}{{A}_{4}}}+...+\frac{{{B}_{n}}{{B}_{1}}}{{{A}_{n}}{{A}_{1}}}>1$$

International Olympiad of Metropolises 2016 P4

A convex quadrilateral ABCD has right angles at A and  C. A point E lies on the extension of the side AD beyond D so that < ABE = < ADC. The point K is symmetric to the point C with respect to point A. Prove that < ADB = < AKE.

International Olympiad of Metropolises 2017 P1

Let ABCD be a parallelogram in which the angle at B is obtuse and AD > AB. Points K and L are chosen on the diagonal AC such that < ABK = < ADL (the points A, K, L, C are all different, with K between A and L). The line BK intersects the circumcircle ω of triangle ABC at points B and E, and the line EL intersects ω at points E and F. Prove that BF // AC.

International Olympiad of Metropolises 2017 P6

Let ABCDEF be a convex hexagon which has an inscribed circle and a circumscribed circle. Denote by ω_A, ω_B, ω_C, ω_D, ω_E, and ω_F the inscribed circles of the triangles FAB, ABC, BCD, CDE, DEF, and EFA, respectively. Let l_AB be the external common  tangent of ω_A and ω_B other than the line AB; lines l_BC, l_CD, l_DE, l_EF , and l_FA are analogously defined. Let A₁ be the intersection point of the lines l_FA and l_AB , B₁ be the intersection point of the lines l_AB and l_BC , points C₁, D₁, E₁, and F₁ are analogously defined. Suppose that A₁B₁C₁D₁E₁F₁ is a convex hexagon. Show that its diagonals A₁D₁, B₁E₁, and C₁F₁ meet at a single point.

International Olympiad of Metropolises 2018 P2

A convex quadrilateral $ABCD$ is circumscribed about a circle $\omega$. Let $PQ$ be the diameter of $\omega$ perpendicular to $AC$. Suppose lines $BP$ and $DQ$ intersect at point $X$, and lines $BQ$ and $DP$ intersect at point $Y$. Show that the points $X$ and $Y$ lie on the line $AC$.

 Géza Kós

International Olympiad of Metropolises 2018 P6

The incircle of a triangle $ABC$ touches the sides $BC$ and $AC$ at points $D$ and $E$, respectively. Suppose $P$ is the point on the shorter arc $DE$ of the incircle such that $\angle APE = \angle DPB$. The segments $AP$ and $BP$ meet the segment $DE$ at points $K$ and $L$, respectively. Prove that $2KL = DE$.

Dušan Djukić

International Olympiad of Metropolises 2019 P3

In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangles $AKL$ lies on the line $IO$.

International Olympiad of Metropolises 2019 P5

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic.

Tibor Bakos and Géza Kós

International Olympiad of Metropolises 2020 P1

In a triangle $ABC$ with a right angle at $C$, the angle bisector $AL$ (where $L$ is on segment $BC$) intersects the altitude $CH$ at point $K$. The bisector of angle $BCH$ intersects segment $AB$ at point $M$. Prove that $CK=ML$

Alexey Doledenok

International Olympiad of Metropolises 2020 P6

In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.

Nairi Sedrakyan and Yuliy Tikhonov

International Olympiad of Metropolises 2021 P2

Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ so that $P$ lies between $B$ and $Q$. The rays $AP$ and $AQ$ divide the angle $BAC$ into three equal parts. It is known that the triangle $APQ$ is acute-angled. Denote by $B_1,P_1,Q_1,C_1$ the projections of points $B,P,Q,C$ onto the lines $AP,AQ,AP,AQ$, respectively. Prove that lines $B_1P_1$ and $C_1Q_1$ meet on line $BC$.

International Olympiad of Metropolises 2021 P6

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

source: megapolis.educom.ru/en