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and 
be 2 distinct positive integers.
be in an integer. Prove why ?
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)
with maximums 
and 
. If 
, and 
can be expressed as 
where 
is a positive integer, find .
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 .
and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
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)
are also the roots of the equation 
. What is the third root of the second equation?
and 
? That wouldn't be a neat numbers to use in a synthetic division. Is there anyway we can do this problem WITHOUT using synthetic division?
Yes.
. From here, like you did, it's not that hard to figure it out.
?
?
are also the roots of the equation . What is the third root of the second equation?
, now use one of the roots for the second equation with synthetic division, I chose 
and then I factored the second equation to: 
.
. Third root is 
. Third root is 
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: .
symbol. Statements imply other statements. An expression on its own cannot imply anything.
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: . symbol. Statements imply other statements. An expression on its own cannot imply anything.
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: . symbol. Statements imply other statements. An expression on its own cannot imply anything. 
. What if someone used 
where they meant 
, or 
where they meant 
. It's not any different to me at least. Anyway, I'm sure your math professors won't like it either, as 10000th User hinted at.
,
.
are 
has 3 roots because it is a cubic.
.
of the roots of the 2nd equation 
)
