MathDB

2

Part of 2009 Turkey Team Selection Test

Problems(2)

Turkey TST Triangle Inequality

Source: Turkey TST 2009, Problem 5

4/5/2009
In a triangle ABC ABC incircle touches the sides AB AB, AC AC and BC BC at C1 C_1, B1 B_1 and A1 A_1 respectively. Prove that \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}} is true.
inequalitiesgeometrygeometry unsolved
Turkey TST 2009 Quadrilateral

Source: Turkey TST 2009, Problem 2

4/5/2009
Quadrilateral ABCD ABCD has an inscribed circle which centered at O O with radius r r. AB AB intersects CD CD at P P; AD AD intersects BC BC at Q Q and the diagonals AC AC and BD BD intersects each other at K K. If the distance from O O to the line PQ PQ is k k, prove that OK\cdot\ k \equal{} r^2.
trigonometrygeometry unsolvedgeometry