[/quote]
,
is tangent to the circumcircle of triangle 
.
respectively, then line 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle 
and 
respectively. Prove that 
be the altitude and 
be the intersection point of the altitudes of triangle 
is symmetric to 
intersects 
.
.
to be a subset of size 
of 
.
such that 
and 
be positive real numbers, such that 
.![\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]](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 
.
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.
that forms an angle of 
or 
with the positive horizontal semi-axis intersects 
points or at no points.
is called good, if 
are non-negative integers, and 
is either 
or 
.
such that 
Suppose Ivan starts with the least possible number of ones to obtain 
after some steps, determine the minimum number of steps he will need in order to do so.
and 
Prove that
Where 
and 
.
Equality holds when 
or 
or 
Y by Lilathebee, ImSh95
p1. Consider the sequence of positive integers 
. which are not perfect squares. Calculate the 
-th term of the sequence.
p2. In a triangle
, let 
be the midpoint of side 
and 
be the midpoint of segment 
. Lines 
and 
intersect at 
. Show that 
.
p3. Find all positive integers
such that 
is a square perfect.
p4. In a party, there is a certain group of people, none of whom has more than
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
. which are not perfect squares. Calculate the 
-th term of the sequence.p2. In a triangle
, let 
be the midpoint of side 
and 
be the midpoint of segment 
. Lines 
and 
intersect at 
. Show that 
.p3. Find all positive integers
such that 
is a square perfect.p4. In a party, there is a certain group of people, none of whom has more than
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?This post has been edited 3 times. Last edited by parmenides51, Sep 4, 2022, 4:14 PM
as the only solution
So,that number is 
.
To get the second sequence we have simply removed the numbers which are perfect squares up to 2019 from sequence 
(which like we said, there are 44), we now know that sequence 
have 2019 elements we need to add 
Notice that after adding those number we have passed 
which is a perfect square,so it must be removed from our sequence.
has 
elements so we must add 
which include all positive integers that are not perfect squares. Prove: for any positive integer n, 
