MathDB

3

Part of 2009 Tuymaada Olympiad

Problems(4)

Prove that angle ABC > 120°

Source: Tuymaada 2009, Junior League, First Day, Problem 3

7/19/2009
In a cyclic quadrilateral ABCD ABCD the sides AB AB and AD AD are equal, CD>AB\plus{}BC. Prove that ∠ABC>120∘ \angle ABC>120^\circ.
trigonometrygeometry unsolvedgeometry
An arrangement of chips in the squares

Source: Tuymaada 2009, Senior League, Second Day, Problem 2

7/19/2009
An arrangement of chips in the squares of n×n n\times n table is called sparse if every 2×2 2\times 2 square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what n n is this possible? Proposed by S. Berlov
combinatorics unsolvedcombinatorics
What is the minimum possible value of AB/CD?

Source: Tuymaada 2009, Senior League, First Day, Problem 3

7/19/2009
On the side AB AB of a cyclic quadrilateral ABCD ABCD there is a point X X such that diagonal BD BD bisects CX CX and diagonal AC AC bisects DX DX. What is the minimum possible value of ABCD AB\over CD? Proposed by S. Berlov
inequalitiesgeometrycyclic quadrilateralgeometry unsolved
Prove that the circumcentre of AB_1C_1 lies on the line AO_1

Source: Tuymaada 2009, Senior League, Second Day, Problem 3

7/19/2009
A triangle ABC ABC is given. Let B1 B_1 be the reflection of B B across the line AC AC, C1 C_1 the reflection of C C across the line AB AB, and O1 O_1 the reflection of the circumcentre of ABC ABC across the line BC BC. Prove that the circumcentre of AB1C1 AB_1C_1 lies on the line AO1 AO_1. Proposed by A. Akopyan
geometry