Olympiad problems

1968 Putnam, B3

Given that a $60^{\circ}$ angle cannot be trisected with ruler and compass, prove that a $\frac{120^{\circ}}{n}$ angle cannot be trisected with ruler and compass for $n=1,2,\ldots$

1997 Tuymaada Olympiad, 4

Tags: geometry , construction , constructions

Using only angle with angle $\frac{\pi}{7}$ and a ruler, constuct angle $\frac{\pi}{14}$

2015 USA Team Selection Test, 1

Tags: functional equation , rational function , algebra , number theory , constructions

Let $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$. [i]Proposed by Victor Wang[/i]

2017 India Regional Mathematical Olympiad, 1

Tags: constructions , geometry

Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on the way $OA$ and $D$ lies on the ray $OB$, and $CP:PD=1:2$.

2011 Sharygin Geometry Olympiad, 5

Tags: geometry , construction , angle bisector , constructions

It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?

2010 Sharygin Geometry Olympiad, 8

Tags: geometry , construction , constructions , concurrency , concurrent

Triangle $ABC$ is inscribed into circle $k$. Points $A_1,B_1, C_1$ on its sides were marked, after this the triangle was erased. Prove that it can be restored uniquely if and only if $AA_1, BB_1$ and $CC_1$ concur.