,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
and 
.
are non-negative real numbers such that 
Prove that![\[\sqrt{a^2+6ab}+\sqrt{b^2+6bc}+\sqrt{c^2+6ca} \ge 5\sqrt{1 + \frac{153abc}{50(a+b+c)}}.\]](http://latex.artofproblemsolving.com/4/2/9/4295e20975f0a7c9902a73695bebb665385f01b3.png)
or 
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and 
.
Y by
Prove that there are no tuples 
sastifying 
sastifying 
This post has been edited 1 time. Last edited by Anabcde, Apr 2, 2025, 1:24 PM
Reason: I forgot the condition that x, y, z must be integers
Reason: I forgot the condition that x, y, z must be integers
