@Kitonum 

evalf fails even for more obvious examples such as

f:=unapply(`if`(n<10^10, 0, 10^10), n);

But adopting this point of view, we should analyse each and every approximation obtained by Maple. Practically this is done only if the result is critical, or if our intuition suspects that something went wrong.

@shahid 

Do yo have an ODE which cannot be solved using the buit-in commands?

The initial version was

A3_2015:=proc(n::posint)
local i;
if type(n,even) then return 0 fi;
add(igcd(i,n),i=1..n);
end:

I'll stop here.

It was a minor edit, ''2''^ was inserted and it was mentioned later.

add(igcd(i, 2014), i = 1 .. 2014);
is never executed. I do not understand your continuous objections.

It should be obvious that A3_2015 computes the product, not its log2 (so that it returns 0 if n is even).

@Markiyan Hirnyk 

13725 is(was) the exponent (power 2) of the answer. I have added the basis to avoid confusion.

@Markiyan Hirnyk 

So, 2015-A3 cannot be solved (as expected) by brute force. One needs:

A3_2015:=proc(n::posint)
local i;
if type(n,even) then return 0 fi;
''2''^add(igcd(i,n),i=1..n);
end:

A3_2015(2015);

(Of course, Maple has a small contribution here.)

@Markiyan Hirnyk 

It should be noted that this is not a true (rigorous) solution of 1999-B5, but only a "guess" done by gfun.

@Earl 

The parametric reduced equation of an ellipse with semiaxes a,b and center O is
x = a cost, y = b sint.

After a rotation with angle u and a translation ==>
x = a cost cosu - b sint sinu + x0, y = a cost sinu + b sint cosu + y0.

The minimum of x when t runs in [0,2Pi] is  - sqrt(a^2 cos^2u + b^2 sin^2u) + x0.
(this follows from dx/dt = 0 etc, or directly using the Cauchy-Schwarz inequality).
This minimum is 0 if the ellipse is tangent to the Oy axis
==> x0 =  sqrt(a^2 cos^2u + b^2 sin^2u).

Similarly for y0.

@Earl 

No, but after a rotation with angle u, the center of the ellipse is

xc = sqrt(a^2*cos(u)^2+b^2*sin(u)^2),
yc = sqrt(a^2*sin(u)^2+b^2*cos(u)^2)

So, xc^2 + yc^2 = a^2 + b^2.

@Markiyan Hirnyk 

This heavily depends on the context and/or definition.

For example, what is (or should be) :

diff(x, f(x)) =  ? 

diff(t, sin(t)) = ?

@hind 

restart;
u := x -> sin(x)+cos(x)-4*x+(alpha-4)*x:
v:=x->sin(x)-cos(x)+beta:
e1:=alpha - int(u(t)+v(t), t = 0 .. Pi):
e2:=beta - int(u(t)-v(t), t = 0 .. Pi):
s:=solve({e1,e2},{alpha,beta});

simplify(eval([e1,e2],s));

   [0,0]

@Christopher2222 

You have solved another system: a=0,b=0  instead of a=alpha,b=beta.

You may set e.g.

infolevel[fsolve]:=2;

to see what fsolve is doing. In your case it finally says "cannot converge to a solution".

You could try to restrict the domain, e.g. {a=0.5 .. 10,  ...}

@asa12 

You did not present your problem. It seems that you want to compute numerically some eigenvectors. But your problem has several parameters, and in order to obtain numerical results you should give numerical values to these parameters.

The expression is too complicated for a symbolic approach. Not to mention that for a symbolic approach it is better to avoid floating-point numbers.