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has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals 
or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
such that 
for all positive integers 
.
+1 w
be a triangle with 
Line bisector of 
intersects 
at point 
. Prove that 
.
. Prove that
Let 
. Prove that
Let 
. Prove that
Let . Prove that
with edges 
.
three perpendiculars to the three lateral faces. from the apex such that these three points lie above the apex.
and radius 
.
. and radius ..
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
, the range being 
?
with edge cm.
and tangent to the plane 
.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
with 
and 
.
and 
are chosen on 
and 
respectively, so that 
and 
.
.