1/(a3(b + c)) + 1/(b3(c + a)) + 1/(c3(a + b)) ≥ 3/2.
xi-1 + 2/xi-1 = 2xi + 1/xi.
| A1. Let A, B, C, D be four distinct points on a line, in that order. The circles with diameter AC and BD intersect at X and Y. The line XY meets BC at Z. Let P be a point on the line XY other than Z. The line CP intersects the circle with diameter AC at C and M, and the line BP intersects the circle with diameter BD at B and N. Prove that the lines AM, DN, XY are concurrent. |
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A2. Let a, b, c be positive real numbers with abc = 1. Prove that:
1/(a3(b + c)) + 1/(b3(c + a)) + 1/(c3(a + b)) ≥ 3/2. |
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| A3. Determine all integers n > 3 for which there exist n points A1, ... , An in the plane, no three collinear, and real numbers r1, ... , rn such that for any distinct i, j, k, the area of the triangle AiAjAk is ri + rj + rk. |
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B1. Find the maximum value of x0 for which there exists a sequence x0, x1, ... , x1995 of positive reals with x0 = x1995 such that for i = 1, ... , 1995:
xi-1 + 2/xi-1 = 2xi + 1/xi. |
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| B2. Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF = FA, such that ∠BCD = ∠EFA = 60o. Suppose that G and H are points in the interior of the hexagon such that ∠AGB = ∠DHE = 120o. Prove that AG + GB + GH + DH + HE ≥ CF. |
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| B3. Let p be an odd prime number. How many p-element subsets A of {1, 2, ... , 2p} are there, the sum of whose elements is divisible by p? |
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