Solutions in math book titled Narrative Approaches to the International Mathematical Problems

is exactly one measure , with 0° ≤  ≤ 90°, for which x = y and that this value of  satisfies < sin < .

Problem 7 of Australia Mathematical Olympiad 2010 On the edges of a triangle ABC are drawn three similar isosceles triangles APB (with AP = PB), AQC (with AQ = QC) and BRC (with BR = RC). The triangles APB and AQC lie outside the triangle ABC and the triangle BRC is lying on the same side of the line BC as the triangle ABC. Prove that the quadrilateral PAQR is a parallelogram.

Problem 5 of Turkey Mathematical Olympiad 2007 Let ABC be a triangle with ∠B = 90°. The incircle of ABC touches the side BC at D. The incenters of triangles ABD and ADC are X and Z, respectively. The lines XZ and AD are intersecting at the point K. XZ and circumcircle of ABC are intersecting at U and V. Let M be the midpoint of line segment [UV]. AD intersects the circumcircle of ABC at Y other than A. Prove that |CY| = 2|MK|.

Problem 2 of Turkey MO Team Selection Test 1996 In a parallelogram ABCD with ∠A < 90°, the circle with diameter AC intersects the lines CB and CD again at E and F, and the tangent to this circle at A meets the line BD at P. Prove that the points P, E, F are collinear.

Problem 6 of Pan African 2009 Points C, E, D and F lie on a circle with center O. Two chords CD and EF intersect at a point N. The tangents at C and D intersect at A, and the tangents at E and F intersect at B. Prove that ON ⊥ AB. Problem 7 of Belarus Mathematical Olympiad 1997 49