The answer is x_1 = ... = x_n = 1/n (n=2014) [actually easy to guess].
The problem is designed for a math competition, not for a CAS, but here is a solution.
The function f : [-1,1] -> R, f(x) = (1+x)^(1/2) is continuous, strictly increasing and strictly concave (f''(x)<0 for |x|<1).
For x_1, ..., x_n in [-1,1], Jensen's inequality implies
f(x_1) + ...+ f(x_n) <= n f((x_1 + ... +x_n)/n).
Denoting x* = (x_1 + ... +x_n)/n, from the 1st equation it results
f(1/n) <= f(x*).
Using the 2nd equation:
f(-1/n) <= f(-x*).
f being strictly increasing, 1/n <= x*, -1/n <= -x*, so, x*=1/n.
It results from the first equation that we have "=" in Jensen's inequqlity.
f being strictly concave, we must have x_1 = ...= x_n = 1/n.