,
,
and 
, then we can deduce 
\le 2 $ ?
and 
.
Equality holds when 
or 
or 
that simultaneously satisfy the conditions 
and![\[
\frac{x^4}{a} - \frac{y^4}{b} = \frac{1}{a - b}.
\]](http://latex.artofproblemsolving.com/4/5/e/45e29c956890cb0bd869c67e7ad94c92d868d2e2.png)
,![\[
\left( \frac{x^2}{a} \right)^n + \left( \frac{y^2}{b} \right)^n = \frac{2}{(a - b)^n}.
\]](http://latex.artofproblemsolving.com/4/0/5/405491f7aae19de4c7ad1f90da3d66e4038e3299.png)
,
,
.
be the angle bisector of angle 
such that 
.
?
of the numbers 
satisfy 
and 
?
for all 
.
with exactly 2 distinct prime factors. Prove that 
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
.
:
is drawn externally on side 
.
is taken on 
.
if 
.
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 
more number to the sequence (for it to have 2019 elements).
which include all positive integers that are not perfect squares. Prove: for any positive integer n, 
