Olympiad problems

1983 Bundeswettbewerb Mathematik, 2

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

1991 Denmark MO - Mohr Contest, 3

Tags: perimeter , right triangle , geometry

A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.

1973 Czech and Slovak Olympiad III A, 1

Tags: geometry , trigonometry , right triangle

Consider a triangle such that \[\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.\] Show that the triangle is right.

2013 India PRMO, 19

Tags: ratio , right triangle , area

In a triangle $ABC$ with $\angle BC A = 90^o$, the perpendicular bisector of $AB$ intersects segments $AB$ and $AC$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $BXYC$ to the area of triangle $ABC$ is $13 : 18$ and $BC = 12$ then what is the length of $AC$?

2006 Sharygin Geometry Olympiad, 10.5

Tags: geometry , tetrahedron , 3d geometry , right triangle

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?

2013 Estonia Team Selection Test, 4

Tags: geometry , cyclic quadrilateral , right triangle , concyclic , square

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

1988 IMO Longlists, 23

Tags: right triangle , geometry , area of a triangle , incenter , ratio

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

2024 Kosovo EGMO Team Selection Test, P3

Tags: geometry , right triangle

Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$. Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.