MathDB

2

Part of 2009 Mexico National Olympiad

Problems(2)

Integers in Boxes

Source: OMM 2009 2

7/15/2014
In boxes labeled 00, 11, 22, …\dots, we place integers according to the following rules:
∙\bullet If pp is a prime number, we place it in box 11.
∙\bullet If aa is placed in box mam_a and bb is placed in box mbm_b, then abab is placed in the box labeled amb+bmaam_b+bm_a.
Find all positive integers nn that are placed in the box labeled nn.
inductionmodular arithmeticnumber theory unsolvednumber theory
A Bunch of Perpendicular Lines

Source: OMM 2009 5

7/15/2014
Consider a triangle ABCABC and a point MM on side BCBC. Let PP be the intersection of the perpendiculars from MM to ABAB and from BB to BCBC, and let QQ be the intersection of the perpendiculars from MM to ACAC and from CC to BCBC. Show that PQPQ is perpendicular to AMAM if and only if MM is the midpoint of BCBC.
geometry unsolvedgeometry