# Blog Archives

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

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

## IMO 2012 problems

This year IMO problems !!!!

Problem 1 :

Given triangle  the point  is the centre of the excircle opposite the vertex  This excircle is tangent to the side  at , and to the lines  and  at  and , respectively. The lines  and  meet at , and the lines  and  meet at  Let  be the point of intersection of the lines  and , and let  be the point of intersection of the lines  and  Prove that  is the midpoint of .

Problem 2 :

Let ${n\ge 3}$ be an integer, and let ${a_2,a_3,\ldots ,a_n}$ be positive real numbers such that  ${a_{2}a_{3}\cdots a_{n}=1}$  Prove that

$\displaystyle \left(a_{2}+1\right)^{2}\left(a_{3}+1\right)^{3}\dots\left(a_{n}+1\right)^{n}>n^{n}.$

Problem 3 :

The liar’s guessing game is a game played between two players ${A}$ and ${B}$. The rules of the game depend on two positive integers ${k}$ and ${n}$ which are known to both players.

At the start of the game ${A}$ chooses integers ${x}$ and ${N}$ with ${1 \le x \le N.}$ Player ${A}$ keeps ${x}$secret, and truthfully tells ${N}$ to player ${B}$. Player ${B}$ now tries to obtain information about ${x}$ by asking player ${A}$ questions as follows: each question consists of ${B}$ specifying an arbitrary set ${S}$ of positive integers (possibly one specified in some previous question), and asking ${A}$whether ${x}$ belongs to ${S}$. Player ${B}$ may ask as many questions as he wishes. After each question, player ${A}$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any ${k+1}$ consecutive answers, at least one answer must be truthful.

After ${B}$ has asked as many questions as he wants, he must specify a set ${X}$ of at most ${n}$positive integers. If ${x}$ belongs to ${X}$, then ${B}$ wins; otherwise, he loses. Prove that:

1. If ${n \ge 2^k,}$ then ${B}$ can guarantee a win. 2. For all sufficiently large ${k}$, there exists an integer ${n \ge (1.99)^k}$ such that ${B}$ cannot guarantee a win.

Problem 4 :

Find all functions ${f:\mathbb Z\rightarrow \mathbb Z}$ such that, for all integers ${a,b,c}$ that satisfy ${a+b+c=0}$, the following equality holds:

$\displaystyle f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$

Problem 5 :

Let ${ABC}$ be a triangle with ${\angle BCA=90^{\circ}}$, and let ${D}$ be the foot of the altitude from ${C}$. Let ${X}$ be a point in the interior of the segment ${CD}$. Let ${K}$ be the point on the segment ${AX}$such that ${BK=BC}$. Similarly, let ${L}$ be the point on the segment ${BX}$ such that ${AL=AC}$. Let ${M}$ be the point of intersection of ${AL}$ and ${BK}$.

Show that ${MK=ML}$.

Problem 6 :

Find all positive integers ${n}$ for which there exist non-negative integers ${a_1, a_2, \ldots, a_n}$ such that

$\displaystyle \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.$