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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