Olympiad problems

2021 Brazil National Olympiad, 6

Let $n \geq 5$ be integer. The convex polygon $P = A_{1} A_{2} \ldots A_{n}$ is bicentric, that is, it has an inscribed and circumscribed circle. Set $A_{i+n}=A_{i}$ to every integer $i$ (that is, all indices are taken modulo $n$). Suppose that for all $i, 1 \leq i \leq n$, the rays $A_{i-1} A_{i}$ and $A_{i+2} A_{i+1}$ meet at the point $B_{i}$. Let $\omega_{i}$ be the circumcircle of $B_{i} A_{i} A_{i+1}$. Prove that there is a circle tangent to all $n$ circles $\omega_{i}$, $1 \leq i \leq n$.

1997 Estonia National Olympiad, 3

Tags: geometry , tetrahedron , volume , inscribed , circumscribed , 3d geometry , sphere

A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

2014 IFYM, Sozopol, 3

Tags: circumscribed , geometry

In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.

Cono Sur Shortlist - geometry, 2018.G5

Tags: geometry , square , inscribed , circumscribed , geometric inequalities , ratio

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

2001 Saint Petersburg Mathematical Olympiad, 9.3

Tags: geometry , pentagon , circumscribed

A convex pentagon $ABCDE$ is given with $AB=BC$, $CD=DE$ and $\angle A=\angle C=\angle E>90^{\circ}$. Prove that the pentagon is circumscribed [I]Proposed by F. Baharev[/i]

1986 Austrian-Polish Competition, 6

Tags: tetrahedron , geometry , concentric , inscribed , circumscribed , 3d geometry , sphere

Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .

2014 German National Olympiad, 6

Tags: geometry , angle , circumscribed , quadrilateral

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

2014 Contests, 3

Tags: circumscribed , geometry

In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.