consecutive positive integers is cool if it contains exactly 
primes. Are there infinitely many cool sets?
Prove that
Let 
Prove that
and 
intersect each other at 
and 
.
and 
respectively. Let 
and 
be the reflection of 
intersect circles 
and 
(both distinct from 
is the second tangent to 
such that 
and 
are both perfect squares.
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Y by
Points 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:12 AM
)
and say it's rational coordinates are (p,q) then move the point to ([p],[q]) where [•] denote the box function and we are done with all points with integral coordinates.
and 
is clearly 1, where the first point is a lattice point and the second one has rational coordinates.
