Problem 4. Let ABC be a triangle with AB=AC. The angle bisectors of ∠CAB and
∠ABC meet the sides BC and CA at D and E, respectively. Let K be the
incenter of triangle ADC. Suppose that ∠BEK=45. Find allpossible values
of ∠CAB
Problem 5. Determine all functions f from the set of positive integers to the
set of positive integers such that, for all positive integers a and b,
there exists a non-degenerate triangle with sides of lengths a, f(b),
and f(b+f(a)-1). (A triangle is non-degenerate if its vertices are not
collinear.)
Problem 6. Let a_1, a_2, ..., a_n be distinct positive integers and let M be
a set of n-1 positive integers not containing s=a_1+a_2+...+a_n. A
grasshopper is to jump along the real axis, starting at the point 0 and
making n jumps to the right with lengths a_1, a_2, ..., a_n in some order.
Prove that the order can be chosen in such a way that the grasshopper
never lands on any points in M.