Olympiad problems

1971 IMO Longlists, 39

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

2022 IMO, 4

Tags: geometry , congruent triangles , parallel , computer problems

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

1986 China Team Selection Test, 1

Tags: angle bisector , trigonometry , perimeter , congruent triangles , geometry

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

1997 Tournament Of Towns, (565) 6

Tags: geometry , combinatorics , combinatorial geometry , congruent triangles

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into n equal segments and the triangle into n congruent triangles. Each of these n triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. (a) If $n =10$, what is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (b)The same question for $n = 9$. (R Zhenodarov)

2020 Adygea Teachers' Geometry Olympiad, 1

Tags: congruent triangles , congruent , geometry

In planimetry, criterions of congruence of triangles with two sides and a larger angle, with two sides and the median drawn to the third side are known. Is it true that two triangles are congruent if they have two sides equal and the height drawn to the third side?

2023 Israel National Olympiad, P3

Tags: geometry , reflection , geometric transformation , congruent triangles

A triangle $ABC$ is given together with an arbitrary circle $\omega$. Let $\alpha$ be the reflection of $\omega$ with respect to $A$, $\beta$ the reflection of $\omega$ with respect to $B$, and $\gamma$ the reflection of $\omega$ with respect to $C$. It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other. Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$, and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$. Prove that the triangles $PQR, ABC$ are congruent.

1941 Moscow Mathematical Olympiad, 073

Tags: geometry , congruent triangles

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.