MathDB

2

Part of 2005 Cono Sur Olympiad

Problems(2)

Cono Sur Olympiad 2005, Problem 2

Source:

8/19/2014
Let ABCABC be an acute-angled triangle and let ANAN, BMBM and CPCP the altitudes with respect to the sides BCBC, CACA and ABAB, respectively. Let RR, SS be the pojections of NN on the sides ABAB, CACA, respectively, and let QQ, WW be the projections of NN on the altitudes BMBM and CPCP, respectively.
(a) Show that RR, QQ, WW, SS are collinear. (b) Show that MP=RSāˆ’QWMP=RS-QW.
geometrycircumcircletrigonometrygeometry proposed
Cono Sur Olympiad 2005, Problem 5

Source:

8/19/2014
We say that a number of 20 digits is special if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
number theory proposednumber theory