[/quote]
,
and 
be positive integers with 
,
of the set of 
”
”
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
of positive integers that satisfy
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of 
,
such that, if we let 
,
we have 
;
is divisible by 
each have a prime factor not dividing 
is irreducible for every natural number 
.
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
be a function such that
for all 
.
.
be real numbers such that 
Prove that 
Where 
.
such that for all 
,
and 
be points on the same circumference with 
.
and 
be the projections of 
and 
,
cuts the segment 
into equal parts.
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
Y by Adventure10, Mathlover_1, OronSH, aidan0626, Blue_banana4, PikaPika999, and 7 other users
Given are 
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
scalene triangles.
.
to be the number of bases of these triangles (we count three bases for each equilateral and one for each nonequilateral isosceles triangle). Clearly 
.
isosceles triangles, yielding that there are at least 
scalene triangles. 
points of the same color and we apply the lemma.
points on the perpendicular bisector). Hence there are at most 
isosceles triangles, so at lesat 
scalene triangles.
points of that color.
isosceles triangles. At the same time there are 
triangles in total, making 130 non-isosceles AKA scalene triangles. 
triangles that can be formed.
be the number of isosceles triangles in the 
to 
is the maximum number of isosceles triangles. Then the minimum number of scalene triangles is 
,
which have 
as a base, since in order for that to happen, 
isosceles triangles, thus at least 
scalene triangles formed from those points.
isosceles triangles and total triangles equal 
so at least 
triangles.
points in the plane (no three collinear), there are atleast ![\[\binom{x}{3}-2\binom{x}2\]](http://latex.artofproblemsolving.com/e/5/e/e5ebd5bd3558da44fe6d1d25dbe24441be359efa.png)
scalene triangles formed by it.
points and the number of scalene triangles it form are atleast ![\[\binom{13}3-2\binom{13}2=130\]](http://latex.artofproblemsolving.com/3/e/4/3e4c8ec22face221b0219029db3c7ffd7171a46d.png)
as desired.
scalene triangles with same colored vertices.
