CZ-SK-MO 2006

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$ ,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$ ,then $b_1=52,a_2=302$ ,and so on). Can $a_7$ be a prime?

Let $m,n$ be positive integers such that the equation (in respect of $x$ ) $$(x+m)(x+n)=x+m+n$$ has at least one integer root. Prove that $\frac{1}{2}n<m<2n$ .

In a scalene triangle $ABC$ ,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively. $I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$ . Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$ .

Given a segment $AB$ in the plane. Let $C$ be another point in the same plane, $H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$ . Find the locus of $M$ for which $A,B,H,I$ are concyclic.

Find all triples $(p,q,r)$ of pairwise distinct primes such that $$p\mid q+r, q\mid r+2p, r\mid p+3q.$$

Find all real solutions $(x,y,z)$ of the system of equations: $$\begin{cases} \tan ^2x+2\cot^22y=1 \tan^2y+2\cot^22z=1 \tan^2z+2\cot^22x=1 \end{cases}$$