Consider a coin with P(Heads) = p (so P(Tails) = 1-p = q.
The probability of exactly 3 heads is P(3 Heads in 5 flips) = 5C3 * p^3 * (1-p)^2 (you want 3 heads and 2 tails).
Use calculus to maximize P(3 Heads in 5 flips) by taking d/dp of P(3 Heads in 5 flips) and solving for p that gives derivative = 0.  (Max must occur where slope = 0, i.e., at the top of the hill the ground is level.)
That gives p = 3/5.  Then you plug that in and find that P(3 Heads in 5 flips) < 1/2.