Olympiad problems

1954 Polish MO Finals, 1

Prove that in an isosceles trapezoid circumscibed around a circle, the segments connecting the points of tangency of opposite sides with the circle pass through the point of intersection of the diagonals.

1957 Moscow Mathematical Olympiad, 346

Tags: geometry , isosceles trapezoid , diagonal , trapezoid

Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.

1992 Tournament Of Towns, (321) 2

Tags: isosceles trapezoid , trapezoid , geometry

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)

1988 Tournament Of Towns, (176) 2

Tags: geometry , diagonal , isosceles trapezoid , parallel , trapezoid

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

1951 Moscow Mathematical Olympiad, 191

Tags: geometry , geometric construction , isosceles trapezoid , trapezoid , isosceles

Given an isosceles trapezoid $ABCD$ and a point $P$. Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$. Note: It is allowed that the vertices of a quadrilateral lie not only not only on the sides of the trapezoid, but also on their extensions.

1978 Czech and Slovak Olympiad III A, 5

Tags: trapezoid , rectangle , isosceles trapezoid , geometry , incenter

Let $ABCS$ be an isosceles trapezoid. Denote $A',B',C',D'$ the incenters of triangles $BCD,CDA,$ $DAB,ABC,$ respectively. Show that $A',B',C',D'$ are vertices of a rectangle.

2023 Novosibirsk Oral Olympiad in Geometry, 4

Tags: trapezoid , geometry , isosceles trapezoid , isosceles

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.

2001 Czech And Slovak Olympiad IIIA, 5

Tags: area , geometry , trapezoid , isosceles trapezoid

A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.

1993 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , isosceles trapezoid , trapezoid , geometric construction , construction , concurrency

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

2005 Thailand Mathematical Olympiad, 1

Tags: geometry , trapezoid , inscribed , isosceles trapezoid

Let $ABCD$ be a trapezoid inscribed in a unit circle with diameter $AB$. If $DC = 4AD$, compute $AD$.

Novosibirsk Oral Geo Oly IX, 2023.4

Tags: geometry , trapezoid , isosceles trapezoid , isosceles

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.