AoPS Academy
sides are colored gold or silver. Prove that there are at least 
different isosceles triangles whose vertices have the same color.
such that 
. Prove that![\[\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq 255\cdot abcd.\]](http://latex.artofproblemsolving.com/d/e/6/de61611f628ea796c706d05941d7ee9b3fc22916.png)
. Find all 
such that for all 
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
that satisfy 
, where 
is a real number. Prove that for all positive , 
be a set of integers satisfying the following property: For every positive integer 
and every set of coefficients 
, all integer roots of the polynomial 
are also elements of . It is given that contains all numbers of the form 
where 
are positive integers. Prove that contains all integers.![$f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$](http://latex.artofproblemsolving.com/8/b/d/8bdc04d75d0654da65463301f6fd0b0af4160b2b.png)
a polynomial whose roots 
are in geometric progression with ration 
. Find 
.
with maximums 
and 
. If 
, and 
can be expressed as 
where 
is a positive integer, find .
and 
the equation 
does not have real roots. Prove that 
.
, every two opposite sides are equal. Let 
denote the point of intersection of 
with the perpendicular bisector of 
. Define 
and 
similarly. Prove that , , and are collinear.
is the midpoint of and 
are the orthogonal projections of 
onto 
From 
we get 
On the other hand, from 
and 
we get
into 
yields 
we conclude that 
are collinear.
be the midpoint of and let the perpendiculars from 
and 
to the line meet it at points 
, respectively. Notice that 
since we are taking projections along , the 
factor cancels out. Notice that 
and 
. Dividing the two relations yields that 
. Hence, multiplying similar relations gives that 
and by the Menelaus' theorem, the points are collinear. and radius 
and analogous ones for 
are coaxial. Any proof along these lines? 
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
