Maths Olympiad Exam

Find the number of ordered triples (x, y, z) of nonnegative integers satisfying the
conditions:
(i) x = y + z;
(ii) x + y + z = 100.

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Maths Olympiad Other Question

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 Suppose P is an interior point of a triangle ABC such that the ratios
d(A,BC)
d(P,BC)
,
d(B,CA)
d(P,CA)
,
d(C,AB)
d(P,AB)