MathDB

2

Part of 2007 CentroAmerican

Problems(2)

ADPE is circumscribable to a circle

Source: Central American Olympiad 2007, Problem 2

6/12/2007
In a triangle ABCABC, the angle bisector of AA and the cevians BDBD and CECE concur at a point PP inside the triangle. Show that the quadrilateral ADPEADPE has an incircle if and only if AB=ACAB=AC.
geometrysymmetryangle bisector
Numbers "ending" in the product of their digits

Source: Central American Olympiad 2007, Problem 5

6/12/2007
Given two non-negative integers m>nm>n, let's say that mm ends in nn if we can get nn by erasing some digits (from left to right) in the decimal representation of mm. For example, 329 ends in 29, and also in 9. Determine how many three-digit numbers end in the product of their digits.