[/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
Two circles were drawn on the plane 
and 
and points 
and 
on them so that the segment 
has the greatest possible length. 
and 
are tangents from point to circle . 
and 
are tangents from point to circle . The circle 
touches the circle internally and the rays and . The circle 
touches the circle internally and the rays and . Prove that the radii of the circles and are equal.
and 
and points 
and 
on them so that the segment 
has the greatest possible length. 
and 
are tangents from point to circle . 
and 
are tangents from point to circle . The circle 
touches the circle internally and the rays and . The circle 
touches the circle internally and the rays and . Prove that the radii of the circles and are equal.This post has been edited 2 times. Last edited by parmenides51, Mar 16, 2021, 8:24 PM
Reason: typo, a_1, a_2 instead of omega_1, omega_2 at the end
Reason: typo, a_1, a_2 instead of omega_1, omega_2 at the end
be the radii of 
,
be the radii of 
.
is tangent to 
and 
meets 
.
which is symmetric, so 
as desired.
