# Blog Archives

## CGMO-2012 (China Girls Math Olympiad 2012) Problem 8

Find the number of integers $k$ in the set $\{0, 1, 2,\cdots, 2012\}$   such that  $\binom{2012}{k}$ is a multiple of $2012$

## CGMO – 2012 ( China Girls Math Olympiad 2012 ) Problem 7

Let  $\{a_n\}$ be a sequence of nondecreasing positive integers such that  $\frac{r}{a_{r}}= k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that  $\frac{s}{a_s} = k$

## CGMO 2012 ( China Girls Math Olympiad 2012) Problem 5

As shown in the figure below, the in-circle of  is tangent to sides  and  at  and  respectively, and  is the circumcenter of . Prove that .

## CGMO-2012 (China Girls Math Olympiad 2012) Problem 4

There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon

## CGMO 2012 (China Girls Math Olympiad 2012)- Problem 2

Circles  and  are tangent to each other externally at . Points  and  are on , lines  and  are tangent to  at  and , respectively, lines  and  meet at point
Prove that

(1) ;
(2) .

## CGMO 2012 (China Girls Math Olympiad 2012) Problem 1

Let  $a_{1}, a_{2},\ldots, a_{n}$ be non-negative real numbers. Prove that                                                                          $\LARGE \frac{1}{1+a_{1}}+\frac{ a_{1}}{(1+a_{1})(1+a_{2})}+\frac{ a_{1}a_{2}}{(1+a_{1})(1+a_{2})(1+a_{3})}\cdots+\frac{ a_{1}a_{2}\cdots a_{n-1}}{(1+a_{1})(1+a_{2})\cdots (1+a_{n})}\le 1.$