be a sequence of positive integers and let 
be the sequence of real numbers given by
Show that, if there exists at least one term among every million consecutive terms of the sequence 
such that 
.
be a positive integer. We start with 
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
be integers greater than 
,
be positive integers not greater than 
.
not greater than ![\[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \]](http://latex.artofproblemsolving.com/7/4/1/7414caaacc9fd761961279f8a37ab6cd6af6b49a.png)
where 
denotes the greatest common divisor of 
.
Y by cubres
An insect starts from 
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
*Too many edits, my brain had gone to a trip
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.*Too many edits, my brain had gone to a trip
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This post has been edited 4 times. Last edited by Rohit-2006, Apr 16, 2025, 2:23 AM
