Olympiad problems

1951 Moscow Mathematical Olympiad, 200

What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)

1999 French Mathematical Olympiad, Problem 3

Tags: ratio , geometry , inradius , geometric inequalities , triangle

For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?

2021 EGMO, 5

Tags: combinatorial geometry , triangle , geometry

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that [list] [*] no three points in $P$ lie on a line and [*] no two points in $P$ lie on a line through the origin. [/list] A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.

Mathley 2014-15, 2

Tags: geometry , triangle , sequence , algebra , area of a triangle , integer

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

2020 AIME Problems, 1

Tags: geometry , triangle

In $\triangle ABC$ with $AB=AC$, point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$, and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$. The degree measure of $\angle ABC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.