[/quote]
,
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than 1. For every group of 6 marked points, he calculated the sum of the numbers on all 15 connecting segments. It turned out that the absolute value of each such sum is at least 
,
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
is given. It is known that triangles 
,
,
,
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle 
and 
with integer coefficients is called \emph{important} if from the divisibility of both differences 
and 
by 
,
and 
are divisible by 100. Does there exist such an important pair of polynomials 
,
,
and 
is also important?
is given. A cube with side length 
is made up of 
unit cubes, each of which is either black or white. It turns out that among any 
cubes that share a common vertex and form a 
cube, there are at most 
black cubes. What is the maximum number of black cubes that could have been used?
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
are positive real numbers, 
then
be positive real numbers with 
.
and 
Prove that
and 
,
,
with cardinality 
,
with 
,
is not a perfect square.
Y by Adventure10, Mango247
Distinct points 
, 
and 
lie on a line in this order. Point 
lies on the perpendicular bisector of the segment 
. Denote by 
the midpoint of the segment . Let 
be the radius of the incircle of the triangle 
and let 
be the radius of the circle with center lying outside the triangle 
, tangent to 
, 
and 
. Prove that 
.
, 
and 
lie on a line in this order. Point 
lies on the perpendicular bisector of the segment 
. Denote by 
the midpoint of the segment . Let 
be the radius of the incircle of the triangle 
and let 
be the radius of the circle with center lying outside the triangle 
, tangent to 
, 
and 
. Prove that 
.
, 
be incircle of 
and A-excircle of 
,
at 
,
,
and 
intersect segment 
at 
.
and 
as desired. 
and A-excircle of 
touch 
and 
,
and 
,
.
