be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
that satisfies: for all different 
,
?
be a triangle with circumcenter 
and internal bisector 
.
again at 
and 
cuts 
.
lie on 
.
at 
respectively. Let circumcircle of 
cuts 
.
.
satisfying
Prove
Y by
(a) Let 
be a monic polynomial so that there exists another real coefficients 
that satisfy
![\[P(x^2-2)=P(x)Q(x)\]](//latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from
(b) For arbitrary polynomial that satisfies (a), determine whether should have real coefficients or not.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from (b) For arbitrary polynomial
that satisfies (a), determine whether should have real coefficients or not.
