CN-SMO 2012

Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$ , such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.

The incircle $I$ of $\triangle ABC$ is tangent to sides $AB,BC,CA$ at $D,E,F$ respectively. Line $EF$ intersects lines $AI,BI,DI$ at $M,N,K$ respectively. Prove that $DM\cdot KE=DN\cdot KF$ .

For composite number $n$ , let $f(n)$ denote the sum of the least three divisors of $n$ , and $g(n)$ the sum of the greatest two divisors of $n$ . Find all composite numbers $n$ , such that $g(n)=(f(n))^m$ ( $m\in N^*$ ).

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$ . Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.

A nonnegative integer $m$ is called a "six-composited number" if $m$ and the sum of its digits are both multiples of $6$ . How many "six-composited numbers" that are less than $2012$ are there?

Find the least natural number $n$ , such that the following inequality holds: $\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$ .

In $\triangle ABC$ , point $D$ lies on side $AC$ such that $\angle ABD=\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\sqrt{3}$ and $AB=1$ , find the size of $\angle AME$ .

Let positive integers $m,n$ satisfy $n=2^m-1$ . $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions: (a) $x, y$ are two arbitrary points in $P_n$ ; (b) starting at point $x$ , the grasshopper leaps $2012$ times and finishes at point $y$ ; (the grasshopper is allowed to travel $x$ and $y$ more than once) (c) there are even number ways for the grasshopper to do (b).