by Jalil_Huseynov, May 17, 2022, 6:44 PM 
of positive integers such that 
is multiple of 
and 
is multiple of 
.by dangerousliri, Jul 19, 2017, 4:31 PM 
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
, the integer 
is nonzero and divides 
.by v_Enhance, May 3, 2013, 8:08 PM for which ![$\dfrac{n^2+1}{[\sqrt{n}]^2+2}$](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 
.by orl, Jul 11, 2012, 10:27 PM 
be an odd integer. Determine all functions 
from the set of integers to itself, such that for all integers 
and 
the difference 
divides 
by Binomial-theorem, Aug 16, 2011, 9:50 PM ![\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\]](http://latex.artofproblemsolving.com/f/d/0/fd0ab11635ee44ad272a54c0f20a2bc47a3596de.png)
cannot all be real if 
.by MithsApprentice, Sep 30, 2005, 8:14 PM 
be a set of integers (not necessarily positive) such that
with 
; and are elements of (possibly equal), then 
also belongs to . is the set of all integers.by orl, Oct 22, 2004, 2:59 PM 
of positive integers such that 
is divisible by 
.
by orl, Sep 28, 2004, 1:11 PM 
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
February 2012
January 2012
December 2011
November 2011
September 2011
August 2011
July 2011
April 2011
March 2011
February 2011
January 2011
December 2010
November 2010
October 2010
September 2010
August 2010
June 2010
May 2010
April 2010
March 2010
February 2010
January 2010
December 2009
November 2009
October 2009
September 2009
August 2009
July 2009
June 2009
May 2009
April 2009
March 2009
February 2009
January 2009
December 2008
November 2008
October 2008
September 2008
August 2008
July 2008
June 2008
May 2008
April 2008
March 2008
February 2008
December 2007
November 2007
October 2007
September 2007
August 2007
July 2007
June 2007
May 2007
April 2007
March 2007
February 2007
December 2006
November 2006
October 2006
September 2006
August 2006
July 2006
June 2006
![\[ a^2b-1 \mid ab^3-1. \]](http://latex.artofproblemsolving.com/7/3/5/735fd61de9a18113846ce2048a00889dcc5ce8a4.png)
be positive integers such that 
is a multiple of 
for all positive integers 
.