Maths Olympiad Exam

Suppose the integers 1, 2, 3, . . . , 10 are split into two disjoint collections a1, a2, a3, a4, a5 and
b1, b2, b3, b4, b5 such that
a1 < a2 < a3 < a4 < a5,
b1 > b2 > b3 > b4 > b5.
(i) Show that the larger number in any pair {aj , bj}, 1 j 5, is at least 6.
(ii) Show that |a1-b1|+|a2-b2|+|a3-b3|+|a4-b4|+|a5-b5| = 25 for every such partition.

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Maths Olympiad Other Question

Let a, b, c be positive integers such that a divides b^2, b divides c^2 and c divides a^2. Prove that abc divides (a + b + c)^7. The circumference of a circle is divided into eight arcs by a convex quadrilateral ABCD, with
four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of
the arcs lying inside the quadrilateral are denoted by p, q, r, s in counter-clockwise direction
starting from some arc. Suppose p + r = q + s. Prove that ABCD is a cyclic quadrilateral.