# Monthly Archives: August 2012

## 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 6

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of  $n$

## 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-3

Find all pairs $(a,b)$ of integers satisfying: there exists an integer  $d \ge 2$ such that $a^n + b^n+1$  is divisible by $d$ for all positive integers $n$

## 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                                                                          $\LARGE \frac{1}{1+a_{1}}+\frac{ a_{1}}{(1+a_{1})(1+a_{2})}+\frac{ a_{1}a_{2}}{(1+a_{1})(1+a_{2})(1+a_{3})}\cdots+\frac{ a_{1}a_{2}\cdots a_{n-1}}{(1+a_{1})(1+a_{2})\cdots (1+a_{n})}\le 1.$

## Functional equations challenge 1

This is a challenging problem !!

Find all continuous  functions      such that for all :

Solution : ( first try it , if not then you can read )

1) replacing the equation by a simpler one :
=============================================
Let     and so   and the equation becomes  and so :

is a continuous function such that
Q.E.D.

2)
=======================

Let
From 1) above, we get that  and so, since  is continuous :
Either , either  for some .

and so

Let then  :  the restriction of  to  :
and so  is a bijection, so is monotonous
implies then  is increasing and so  is increasing.

So,    and so  which is impossible and so
Same,    and so  which is impossible and so

So
Q.E.D.

3) general solution of  with  continuous
===========================================================

Then the original equation  becomes  (since ) and so :

The general continuous solutions of  are any odd continuous function  such that
A simple way to build them is :
3.1) If  :

3.2) If  for some  :
Let any continuous fonction  from  such that  and define  as :
:
:
:

Notice that  gives the solution

4) general solution of  with  continuous
=============================================================
The general solution of original equation is  where  is any function described in 3) above

5) examples of solutions
======================
gives the solution

gives the solution

and  gives the solution :
If  :
If  :

and  gives the solution :

Notice that the only differentiable solutions are  and