[/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 
.
b) 
b) 
Y by
Given the cube 
. The point 
is an inner point of the edge 
such that 
. Through the points and 
a plane 
is drawn , parallel to the line 
.
a) Denote
the point of intersection of the plane with the line 
. Find the value of the ratio 
.
b) The plane divides the cube into 
parts. Find the ratio of the volume of these parts.
. The point 
is an inner point of the edge 
such that 
. Through the points and 
a plane 
is drawn , parallel to the line 
.a) Denote
the point of intersection of the plane with the line 
. Find the value of the ratio 
.b) The plane
divides the cube into 
parts. Find the ratio of the volume of these parts.This post has been edited 1 time. Last edited by parmenides51, Mar 18, 2021, 12:40 AM
