Olympiad problems

2007 Postal Coaching, 4

Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ of a triangle $ABC$ whose incentre is $I$. Suppose $EF$, extended, meets the circumcircle of $ABC$ in $M,N$. Show that the circumradius of $MIN$ is twice that of $ABC$.

2004 Thailand Mathematical Olympiad, 21

Tags: geometry , inradius , circumradius

The ratio between the circumradius and the inradius of a given triangle is $7 : 2$. If the length of two sides of the triangle are $3$ and $7$, and the length of the remaining side is also an integer, what is the length of the remaining side?

1985 All Soviet Union Mathematical Olympiad, 408

Tags: circumcircle , geometry , arc , circumradius

The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points. Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius.

1983 Bundeswettbewerb Mathematik, 2

Tags: geometry , inradius , circumradius , right triangle , construction

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.