MathDB

2

Part of 2007 International Zhautykov Olympiad

Problems(2)

convex quadrilateral suggesting a trigonometric solution

Source: Third Zhautykov Olympiad, Kazakhstan, 2007

4/25/2007
Let ABCDABCD be a convex quadrilateral, with ∠BAC=∠DAC\angle BAC=\angle DAC and MM a point inside such that ∠MBA=∠MCD\angle MBA=\angle MCD and ∠MBC=∠MDC\angle MBC=\angle MDC. Show that the angle ∠ADC\angle ADC is equal to ∠BMC\angle BMC or ∠AMB\angle AMB.
trigonometrygeometrygeometric transformationreflectiongeometry proposed
a partition in three non-empty subsets...

Source: Third Zhautykov Olympiad, Kazakhstan, 2007

4/25/2007
The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets (A∪B∪C)(A\cup B\cup C). a) show that there exists a triangle of side-lengths a,b,ca,b,c, such that a∈A,b∈B,c∈Ca\in A, b\in B, c\in C. b) does it always happen that there exists a right triangle with the above property ?
inductioncombinatorics proposedcombinatorics