black and 
red cards. You pull out 
cards, one after another, and check their colour. If both cards are the same colour, then a black card is added to the deck. However, if the cards are of different colours, then a red card is used to replace them. Once the cards are taken out of the deck, they are not returned to the deck, and thus the number of cards keeps reducing. What is the likelihood the last card left in the deck is black?
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
Y by Adventure10
Let 
be a triangle and 
be the feet of the altitudes from 
. Let 
respectively 
, be the orthogonal projection of 
onto 
, respectively 
, points 
and 
are defined in an analogous way. The lines 
and 
meet at 
, the lines and 
meet at 
, while the lines and meet at 
. Show that the lines 
and 
are concurrent.
be a triangle and 
be the feet of the altitudes from 
. Let 
respectively 
, be the orthogonal projection of 
onto 
, respectively 
, points 
and 
are defined in an analogous way. The lines 
and 
meet at 
, the lines and 
meet at 
, while the lines and meet at 
. Show that the lines 
and 
are concurrent.
.
or 
pass through symmedian point of 
.
