Asian Pacific Mathematics Olympiad · 2013
Problems
- Problem 1Let ABC be an acute triangle with altitudes AD, BE, and CF, and let O be the center of its circumcircle. Show that the segments OA, OF, OB, OD, OC, OE dissect triangle ABC into three pairs of triangles having equal areas.Solutions: 1
- Problem 2Determine all positive integers n for which (n^2+1)/(floor(sqrt(n))^2+2) is an integer. Here floor(r) denotes the greatest integer less than or equal to r.Solutions: 1
- Problem 3For 2k real numbers a_1, ..., a_k, b_1, ..., b_k, define X_n=sum from i=1 to k of floor(a_i n+b_i), for n=1,2,... . If the sequence X_n is an arithmetic progression, prove that sum from i=1 to k of a_i is an integer. Here floor(r) denotes the greatest integer less than or equal to r.Solutions: 1
- Problem 4Let a and b be positive integers, and let A and B be finite sets of integers satisfying: (i) A and B are disjoint; (ii) if an integer i belongs either to A or to B, then i+a belongs to A or i-b belongs to B. Prove that a|A|=b|B|, where |X| denotes the number of elements of X.Solutions: 1
- Problem 5Let ABCD be a quadrilateral inscribed in a circle omega, and let P be a point on the extension of AC such that PB and PD are tangent to omega. The tangent at C intersects PD at Q and the line AD at R. Let E be the second point of intersection of AQ and omega. Prove that B, E, R are collinear.Solutions: 1