CMO 2014

Let $ABC$ be a triangle with $AB>AC$ . Let $D$ be the foot of the internal angle bisector of $A$ . Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$ .

For the integer $n>1$ , define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$ . Prove that for any integer $k>1$ , there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$ .

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$ ; ii) $f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$ . For each integer $m\ge 2$ , find the value of $f(2^m)$ .

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$ . Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$ . Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$ , there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$ .

Let $f:X\rightarrow X$ , where $X=\{1,2,\ldots ,100\}$ , be a function satisfying:

1. $f(x)\neq x$ for all $x=1,2,\ldots,100$ ;
2. for any subset $A$ of $X$ such that $|A|=40$ , we have $A\cap f(A)\neq\emptyset$ . Find the minimum $k$ such that for any such function $f$ , there exist a subset $B$ of $X$ , where $|B|=k$ , such that $B\cup f(B)=X$ .

For non-empty number sets $S, T$ , define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$ . Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$ . Show that there exists a subset $D$ of $A+B$ such that

1. $D+D\subseteq 2(A+B)$ ,
2. $|D|\geq\frac{|A|\cdot|B|}{2n}$ , where $|X|$ is the number of elements of the finite set $X$ .