2011/065) If sides of length a b c form a triangle prove sqrt(a), sqrt(b), sqrt(c) form a triangle

Let's assume a >= b >= c > 0. As a, b, c are sides of a triangle, they fit the triangle inequality a < b + c.

(1) (sqrt(b) + sqrt(c))^2 = b + 2*sqrt(b)*sqrt(c) + c

Value of sqrt(x) is positive for any positive x, so

(2) (sqrt(b) + sqrt(c))^2 > b + c > a

As square root is an increasing function, (2) follows

(3) sqrt(b) + sqrt(c) > sqrt(a)

Hence proved