Olympiad problems

2020 Adygea Teachers' Geometry Olympiad, 1

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?

congruent triangles geometry congruent number theory

1949-56 Chisinau City MO, 26

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

congruent triangles geometry congruent

2015 Oral Moscow Geometry Olympiad, 1

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

equal angles geometry trapezoid congruent diagonal

2000 Bosnia and Herzegovina Team Selection Test, 5

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

perimeter geometry congruent inequalities number theory

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?

geometry polygon interior congruent polygons

2004 Junior Balkan Team Selection Tests - Romania, 1

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]

circumcircle equal angles geometry incircle congruent number theory

2000 Czech And Slovak Olympiad IIIA, 2

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.

equal circles geometry isosceles incircle congruent number theory

Kyiv City MO Seniors 2003+ geometry, 2010.11.3

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)

geometry inscribed orthocenter congruent complex numbers

2015 Costa Rica - Final Round, G5

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 $PQ \parallel AB$.

geometry parallel congruent number theory

2021 Oral Moscow Geometry Olympiad, 2

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

perimeter geometry congruent

1993 Czech And Slovak Olympiad IIIA, 6

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

tetrahedron 3d geometry combinatorial geometry congruent similar number theory solid geometry

1997 Bundeswettbewerb Mathematik, 3

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?

geometry square congruent number theory

2020 Francophone Mathematical Olympiad, 1

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.

excircle circumcircle congruent triangles geometry congruent number theory spiral similarity

2008 Junior Balkan Team Selection Tests - Moldova, 3

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 $\angle QOP = \frac12 \angle B_1C_1D_1$.

geometry angle rhombus congruent