AoPS Academy
be a triangle. 
is the midpoint of segment 
, and points 
, 
are selected on sides 
, 
respectively such that , , are collinear. The circumcircles 
and 
intersect at a point 
. The circumcircle 
intersects line again at a point 
. Show that the lines 
, 
and the tangent to at point 
concur.
such that 
for all 
people playing a game at Cheesy's math casino, where is an odd prime number. Let 
be a positive integer. A subset of length 
from the set of integers from 
to inclusive is randomly chosen, with an equal probability (
and is fixed). The winner of Cheesy's game is person 
, if the sum of the chosen numbers are congruent to 
for 
., find all values of such that no one will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).
. 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.
be a point lying on the Neuberg cubic of 
and let 
be the circumcenter of 
respectively. Prove that the reflection of 
in 
respectively are concurrent.
be a point lying on the Neuberg cubic of and let be the circumcenter of respectively. Prove that the reflection of in respectively are concurrent.
