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has how many digits?
be a triangle with orthocenter 
and let 
be the second intersection of the circumcircle of triangle 
with the internal bisector of the angle 
. Let 
be the circumcenter of triangle 
and 
the orthocenter of triangle 
. Prove that the length of segment 
is equal to the circumradius of triangle .
such that the following property holds: if we list the divisors of 
in increasing order as 
, then we have![\[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \]](http://latex.artofproblemsolving.com/a/d/c/adc94af88b3579164aabba51bd021442133e364a.png)
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
be the incircle of a fixed equilateral triangle . Let 
be a variable line that is tangent to and meets the interior of segments 
and 
at points and 
, respectively. A point 
is chosen such that 
and 
. Find all possible locations of the point , over all choices of .
be positive real numbers such that ![\(a+b+c=4\sqrt[3]{abc}\)](http://latex.artofproblemsolving.com/2/8/2/28214a1814762b71858db1543fae1a50212eaafa.png)
. Prove that ![\[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]](http://latex.artofproblemsolving.com/7/0/4/7043a170ac5535465a5a1b4daccb1f1de3043280.png)
are multiples of 
or 
but not 
?
![\[222{,}222-22{,}222-2{,}222-222-22-2?\]](http://latex.artofproblemsolving.com/1/5/6/156c3662fe4c874a80e54887f76c302c13b9d14c.png)
