IMO 2014 – Day 1

1. Let a_0 < a_1 < a_2 \ldots be an infinite sequence of positive integers. Prove that there exists a unique integer n\geq 1 such that
a_n < \dfrac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.

2. Let n \ge 2 be an integer. Consider an n \times n chessboard 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 configuration of n rooks, there is a k \times k square which does not contain a rook on any of its k^2 unit squares.

3. Convex quadrilateral ABCD has \angle ABC = \angle CDA = 90^{\circ}. Point H is the foot of the perpendicular from A to BD. Points S and T lie on sides AB and AD, respectively, such that H lies inside triangle SCT and \angle CHS - \angle CSB = 90^{\circ},

\angle THC - \angle DTC = 90^{\circ}. Prove that line BD is tangent to the circumcircle of triangle TSH.

6 thoughts on “IMO 2014 – Day 1”

      1. Thầy cho là em tự lập bảng với n bằng vài giá trị rồi. Nếu đoán vậy thì chứng minh với n bằng m.m cộng 1 xem? Nhớ là phải đủ HIỂU bài toán nhé!

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