Maths Olympiad Exam

Let a, b, c be three positive real numbers such that a + b + c = 1. Prove that among the three numbers a - ab, b - bc, c - ca there is one which is at most 1/4 and there is one which is at least 2/9

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Maths Olympiad Other Question

If n is an integer greater than 7, prove that
n^7-hn^7 is divisible by 7. Here n^7 denotes the number of ways of choosing 7 objects from among n objects; also, for any real number x, [x] denotes the greatest integer not exceeding x.
Find the number of ordered triples (x, y, z) of nonnegative integers satisfying the
conditions:
(i) x = y + z;
(ii) x + y + z = 100.