Olympiad problems

2012 Oral Moscow Geometry Olympiad, 2

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

2021 Oral Moscow Geometry Olympiad, 2

Tags: perimeter , congruent , geometry

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?

1979 Austrian-Polish Competition, 5

Tags: 3d geometry , circumcenter , congruent , tetrahedron , geometry , incenter

The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.

2008 Junior Balkan Team Selection Tests - Moldova, 3

Tags: rhombus , congruent , geometry , angle

Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side $BC$ intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if $\angle QOP = \frac12 \angle B_1C_1D_1$.

1993 Czech And Slovak Olympiad IIIA, 6

Tags: 3d geometry , similar , congruent , combinatorial geometry , tetrahedron

Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.

1949-56 Chisinau City MO, 26

Tags: congruent triangles , congruent , geometry

Formulate a criterion for the conguence of triangles by two medians and an altitude.

2015 Costa Rica - Final Round, G5

Tags: parallel , congruent , geometry

Let $A, B, C, D$ be points that lie on the same circle . Let $F$ be such that the arc $AF$ is congruent with the arc $BF$. Let $P$ be the intersection point of the segments $DF$ and $AC$. Let $Q$ be intersection point of the $CF$ and $BD$ segments. Prove that $PQ \parallel AB$.

2004 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , congruent , equal angles , incircle , circumcircle

Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent. Valentin Vornicu [hide= about Romania JBMO TST 2004 in aops]I found the Romania JBMO TST 2004 links [url=https://artofproblemsolving.com/community/c6h5462p17656]here [/url] but they were inactive. So I am asking for solution for the only geo I couldn't find using search. The problems were found [url=https://artofproblemsolving.com/community/c6h5135p16284]here[/url].[/hide]

2015 Oral Moscow Geometry Olympiad, 1

Tags: equal angles , trapezoid , geometry , diagonal , congruent

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

2000 Bosnia and Herzegovina Team Selection Test, 5

Tags: geometry , perimeter , congruent

Let $T_m$ be a number of non-congruent triangles which perimeter is $m$ and all its sides are positive integers. Prove that: $a)$ $T_{1999} > T_{2000}$ $b)$ $T_{4n+1}=T_{4n-2}+n$, $(n \in \mathbb{N})$

1997 Bundeswettbewerb Mathematik, 3

Tags: geometry , congruent , square

A square $S_a$ is inscribed in an acute-angled triangle $ABC$ with two vertices on side $BC$ and one on each of sides $AC$ and $AB$. Squares $S_b$ and $S_c$ are analogously inscribed in the triangle. For which triangles are the squares $S_a,S_b$, and $S_c$ congruent?

2020 Francophone Mathematical Olympiad, 1

Tags: geometry , circumcircle , excircle , congruent triangles , congruent

Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$ and $X$ the point of $\Gamma$ such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.

2012 Tournament of Towns, 2

Tags: geometry , combinatorial geometry , convex polyhedron , regular polygon , congruent

Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?

1990 All Soviet Union Mathematical Olympiad, 514

Tags: rectangle , geometry , tiling , congruent , square , combinatorial geometry

Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?

2000 Czech And Slovak Olympiad IIIA, 2

Tags: isosceles , equal circles , congruent , incircle , geometry

Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where $E$ and $F$ are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.

Kyiv City MO Seniors 2003+ geometry, 2010.11.3

Tags: congruent , inscribed , orthocenter , geometry

The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)

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?

2019 Oral Moscow Geometry Olympiad, 2

Tags: geometry , diagonal , congruent , quadrilateral , congruence

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?