▪ Asian Pacific Mathematical Olympiad 2013

1. Let be an acute triangle with altitudes , , and , and let be the center of its circumcircle. Show that the segments , , , , , dissect the triangle into three pairs of triangles that have equal areas. 

2. Determine all positive integers for which 

 

is an integer. Here denotes the greatest integer less than or equal to . 

3. For real numbers , define a sequence of numbers by

If the sequence forms an arithmetic progression, show that must be an integer. Here denotes the greatest integer less than or equal to . 

4. Let and be positive integers, and let and be finite sets of integers satisfying

(i) and are disjoint;

(ii) if an integer belongs to either to or to , then either belongs to or belongs to .

Prove that . (Here denotes the number of elements in the set .) 

5. Let be a quadrilateral inscribed in a circle , and let be a point on the extension of such that and are tangent to . The tangent at intersects at and the line at . Let be the second point of intersection between and . Prove that , , are collinear.

Πηγή: artofproblemsolving