[/quote]
,
and 
,![\[0<\frac{a-b}{1+ab} < \sqrt{2}-1.\]](http://latex.artofproblemsolving.com/1/3/f/13fde7ce3103520261558d05100032a735802cbc.png)
be positive integers such that 
and 
.
be the set of values attained by 
as 
runs through the positive integers. Show that 
with 
;
and 
are elements of 
also belongs to 
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
is nonzero and divides 
.
of positive integers such that 
is divisible by 
.
be an odd integer. Determine all functions 
from the set of integers to itself, such that for all integers 
divides 
for which there exist infinitely many positive integers ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
such that ![\[ a^2b-1 \mid ab^3-1. \]](http://latex.artofproblemsolving.com/7/3/5/735fd61de9a18113846ce2048a00889dcc5ce8a4.png)
is a multiple of 
for all positive integers 
.![$\dfrac{n^2+1}{[\sqrt{n}]^2+2}$](http://latex.artofproblemsolving.com/7/e/d/7ed6c67114c0c7d262f67e5b7a733daab7997415.png)
is an integer. Here ![$[r]$](http://latex.artofproblemsolving.com/2/f/c/2fcabe2c2b8ebe2c428830acb9874cb8a8df790f.png)
denotes the greatest integer less than or equal to 
.![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
is multiple of 
and 
is multiple of 
.
and 
where each summation runs over 
variables 
and 
Then find the value of 
Y by
The rectangle 
represents the A4 sheet, that is 
. The rectangle (the paper) is folded such that after the folding the point 
coincides with the point 
on the side 
and the broken line 
runs along the point 
. The length of the side 
is 
. Find the sidelengths of the triangle 
.
represents the A4 sheet, that is 
. The rectangle (the paper) is folded such that after the folding the point 
coincides with the point 
on the side 
and the broken line 
runs along the point 
. The length of the side 
is 
. Find the sidelengths of the triangle 
.This post has been edited 3 times. Last edited by parmenides51, May 30, 2021, 8:45 AM
.
cuts 
in the point 
.
of 
.
cuts 
in the point 
.
