Something Different!

1)Let p be an odd prime and n a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length p^n. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are  integers  divisible by p^n+1


2)Show that 1+2+3+........+n divides 1^k+2^k+...........+n^k, for and odd positive integer k and for nay natural number n


3)Denote by a, b, c the lengths of the sides of a triangle. Prove that
a^2(b + c − a) + b^2(c + a − b) + c^2(a + b − c) ≤ 3abc.

4)Prove that the 8th power of any integer is of the form 17k or 17k+1 or 17k-1 where k is an integer #Maths #Number Theory

5) From ten distinct two-digit numbers, one can always choose two-disjoint nonempty subsets, so that their elemets have the same sum #Maths #IMO 1972