[/quote]
,
and a polynomial 
with real coefficients of degree 
)![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
and 
,
,
the pair 
if for any elements 
we have 
.
be the greatest possible number of elements of a set that escapes the pair 
.
with incircle 
,
be the incenter and 
be the point where 
.
be the point on 
with 
and 
be the orthocenter of 
,
on 
also satisfies 
.
,
,
satisfying 
the following inequality holds:
of the triangle 
.
the orthogonal projection of 
on 
.
and 
are such that 
is a paralellogram, find the locus of point 
when 
.
be the circle tangent to 
passing through 
.
be the circle tangent to 
be the circle tangent to 
passing through 
.
its circumcenter and 
its circumcircle. Let 
and 
be points on 
,
.
,
.
.
and the tangent to 
are concurrent and that 
.
is tangent to the circumcircle of triangle 
and line 
intersect 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle 
respectively. Prove that 
of 
.
pairwise disjoint, non-empty subsets 
such that 
and 
.![\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]](http://latex.artofproblemsolving.com/e/9/d/e9d9385c60d7f3d0854fe16687e0361713579740.png)
,
be the maximum number of points inside or on the sides of a regular 
such that the distance between any two points is greater than 
for 
.
Y by
Let 
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
If is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.If
is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.This post has been edited 1 time. Last edited by Anto0110, Mar 18, 2025, 7:37 PM
Reason: 9
Reason: 9
into sets which can be computed as 
by A000110, and this 
work.
