[/quote]
,
that satisfy 
are positive real numbers, then
and 
Prove that
such that 
is a perfect square for all integers 
and 
.
Prove that
Prove that
and 
Prove that
satisfied 
Find minimum and maximum of 
such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
Proposed by Liam Baker, South Africa
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
Y by Adventure10
The circle inscribed in the triangle 
, with center 
, touches the sides 
and 
at the points 
and 
, respectively. The lines 
and 
intersect the line 
at the points 
and 
. 
is the midpoint of 
and 
. Prove that 
is a regular triangle.
, with center 
, touches the sides 
and 
at the points 
and 
, respectively. The lines 
and 
intersect the line 
at the points 
and 
. 
is the midpoint of 
and 
. Prove that 
is a regular triangle.This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2022, 2:30 AM
on the circle centered at 
with radius 
.
,
.
,
