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. --

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