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2016年第8届罗马尼亚大师赛第二天试题


The 8th Romanian Master of Mathematics Competition
Day 2: Saturday, February 27, 2016, Bucharest

Language: English Problem 4. Let x and y be positive real numbers such that x

+ y 2016 ≥ 1. Prove that x2016 + y > 1 ? 1/100. Problem 5. A convex hexagon A1 B1 A2 B2 A3 B3 is inscribed in a circle ? of radius R. The diagonals A1 B2 , A2 B3 , and A3 B1 concur at X . For i = 1, 2, 3, let ωi be the circle tangent to the segments XAi and XBi , and to the arc Ai Bi of ? not containing other vertices of the hexagon; let ri be the radius of ωi . (a) Prove that R ≥ r1 + r2 + r3 . (b) If R = r1 + r2 + r3 , prove that the six points where the circles ωi touch the diagonals A1 B2 , A2 B3 , A3 B1 are concyclic. Problem 6. A set of n points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets A and B . An AB tree is a con?guration of n ? 1 segments, each of which has an endpoint in A and the other in B , and such that no segments form a closed polyline. An AB -tree is transformed into another as follows: choose three distinct segments A1 B1 , B1 A2 and A2 B2 in the AB -tree such that A1 is in A and A1 B1 + A2 B2 > A1 B2 + A2 B1 , and remove the segment A1 B1 to replace it by the segment A1 B2 . Given any AB -tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after ?nitely many steps.

Each of the three problems is worth 7 points. 1 Time allowed 4 2 hours.