f(x(f(y)) = yf(x) for all x, y; and f(x) → 0 as x → ∞.
a2b(a - b) + b2c(b - c) + c2a(c - a) ≥ 0.
Determine when equality occurs.
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A1. Find all functions f defined on the set of positive reals which take positive real values and satisfy:
f(x(f(y)) = yf(x) for all x, y; and f(x) → 0 as x → ∞. |
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| A2. Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1 and O2 respectively. One of the common tangents to the circles touches C1 at P1 and C2 at P2, while the other touches C1 at Q1 and C2 at Q2. Let M1 be the midpoint of P1Q1 and M2 the midpoint of P2Q2. Prove that ∠O1AO2 = ∠M1AM2. |
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| A3. Let a , b and c be positive integers, no two of which have a common divisor greater than 1. Show that 2abc - ab - bc - ca is the largest integer which cannot be expressed in the form xbc + yca + zab, where x, y, z are non-negative integers. |
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| B1. Let ABC be an equilateral triangle and E the set of all points contained in the three segments AB, BC and CA (including A, B and C). Determine whether, for every partition of E into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. |
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| B2. Is it possible to choose 1983 distinct positive integers, all less than or equal to 105, no three of which are consecutive terms of an arithmetic progression? |
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B3. Let a, b and c be the lengths of the sides of a triangle. Prove that
a2b(a - b) + b2c(b - c) + c2a(c - a) ≥ 0. Determine when equality occurs. |
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