Maths Olympiad
Exam
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.
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Maths Olympiad Other Question
Solve the following equation for real x:
(x^2 + x - 2)^3 + (2x^2 - x - 1)^3 = 27(x^2 - 1)^3
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.