1. Let a_0 < a_1 < a_2 ... be an infinte sequence of positive integers. Prove that there exists a unique integer n≧1 such that a_0 + a_1 + a_2 + ... + a_n a_n < ---------------------------- ≦ a_n+1 n 2. Let n≧2 be an integer. Conisder an n ×n chesboard consisting of n^2 unit squares. A configuration of n rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer k such that, for each peaceful configuartion of n rooks, there is a k ×k square which does not contain a rook on any of its k^2 unit squares. 3. Convex quadrilateral ABCD has ∠ABC = ∠CDA = 90. Point H is the foot of the perpendicular A to BD. Point S and T lie on sides AB and AD, respectively, such that H lies inside triangle SCT and ∠CHS - ∠CSB = 90 , ∠THC - ∠DTC = 90 Prove that line BD is tangent to the circumcircle of triangle TSH. --

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