Asian Pacific Mathematics Olympiad · 1999
Problems
- Problem 1Find the smallest positive integer n such that no arithmetic progression of 1999 real terms contains exactly n integers.Solutions: 1
- Problem 2Let a1,a2,... be a sequence of real numbers satisfying a(i+j) <= ai+aj for all positive integers i,j. Prove that a1+a2/2+...+an/n >= an for every positive integer n.Solutions: 1
- Problem 3Two circles touch the line AB at A and B and intersect at X and Y, with X nearer to AB. The tangent to the circle AXY at X meets the circle BXY at W. The ray AX meets BW at Z. Prove that BW and BX are tangents to the circle XYZ.Solutions: 1
- Problem 4Find all pairs of integers (m,n) such that m squared plus 4n and n squared plus 4m are both perfect squares.Solutions: 1
- Problem 5A set of 2n+1 points in the plane has no three collinear and no four concyclic. A circle divides the set if it passes through 3 of the points and has exactly n-1 points inside it. Prove that the number of circles dividing the set is even if and only if n is even.Solutions: 1