Question

What is the maximum angle  at which you can throw a ball so that its distance from you never decreases during its flight?

Answer 1

There may be more elaborate methods, but the following should work:
Use kinematic equations:
u=initial velocity (m/s)
\(a=initial angle with horizontal (deg)\)
t=elapsed time (s)
g=-9.8 m/s^2 = acceleration due to gravity
then
x=u cos(a)t
y=u sin(a)t+(1/2)gt^2

Distance from original position
d(t)=sqrt(x^2+y^2)

Square of distance with respect to time
D(t) = d^2=x^2+y^2

Evaluate D(t) and find "a" such that D'(t) is increasing for t>0.