Olympiad problems

1953 Putnam, B6

Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.

2009 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: board , minimal , combinatorics

There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board. $a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board $b)$ For such $n$, find numbers written on the board at the beginning

2009 Romania National Olympiad, 2

Tags: complex number , minimal , complex disc , inequalities

Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $

2011 Sharygin Geometry Olympiad, 24

Tags: geometry , minimal , sides of a triangle

Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

2013 Oral Moscow Geometry Olympiad, 6

Tags: minimum , circumcircle , geometry , distance , minimal , circumcenter

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point $ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.