MathDB

4

Part of 2003 Balkan MO

Problems(1)

balkan problem

Source: Balkan Math Olympiad BkMO 2003, problem 4

4/25/2004
A rectangle ABCDABCD has side lengths AB=mAB = m, AD=nAD = n, with mm and nn relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points A1=A,A2,A3,…,Ak=CA_1 = A, A_2, A_3, \ldots , A_k = C. Show that A1A2−A2A3+A3A4−⋯+Ak−1Ak=m2+n2mn. A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}.
geometryrectanglesearchnumber theoryrelatively primecombinatorics unsolvedcombinatorics