Olympiad problems

2021 Sharygin Geometry Olympiad, 8.8

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

1966 IMO Longlists, 41

Tags: geometry , polygon , counting , obtuse triangle

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

2010 IMAC Arhimede, 5

Tags: combinatorial geometry , point , obtuse triangle , combinatorics

Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$

May Olympiad L1 - geometry, 2019.4

Tags: paper , cutting the paper , obtuse triangle , geometry , square , combinatorics

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

2010 Saudi Arabia IMO TST, 1

Tags: combinatorics , geometry , combinatorial geometry , obtuse triangle

Let $A_1A_2...A_{2010}$ be a regular $2010$-gon. Find the number of obtuse triangles whose vertices are among $A_1$, $A_2$,$ ...$, $A_{2010}$.

1966 IMO Shortlist, 41

Tags: geometry , polygon , counting , obtuse triangle

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

2019 May Olympiad, 4

Tags: combinatorics , geometry , cutting the paper , paper , obtuse triangle , square

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

2008 Bulgarian Autumn Math Competition, Problem 8.4

Tags: geoemtry , obtuse triangle , combinatorics

Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?

2015 Thailand Mathematical Olympiad, 4

Tags: obtuse triangle , concyclic , incircle , geometry

Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.

1967 German National Olympiad, 4

Tags: geometry , polygon , counting , obtuse triangle

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?