**A. 567.** (*a*) Find all pairs (*a*,*b*) of relatively prime positive integers *a*, *b* such that *b* divides *a*^{2}-3 and *a* divides *b*^{2}-3. (*b*) Find all pairs (*a*,*b*) of relatively prime positive integers *a*, *b* such that *b* divides *a*^{2}-5 and *a *divides *b*^{2}-5.

Proposed by: *J. Pelikán,* Budapest

(5 points)

**A. 568.** Given a triangle *ABC* and a line through its incenter. Denote by *A*', *B*' and *C*' the mirror images of *A*, *B* and *C* about , respectively. Let the lines through *A*', *B*' and *C*', parallel to , meet the lines *BC*, *CA* and *AB* at *P*, *Q* and *R*, respectively. Prove that the points *P*, *Q* and *R* lie on a line and this line is tangent to the incircle.

(5 points)

**C. 1133.** The digits of the sixth power of a natural number in ascending order are as follows: 0, 2, 3, 4, 4, 7, 8, 8, 9. Which number is it?

(5 points)