Observe the following equalities:
sqrt(49) = 4 + sqrt(9)
sqrt(64) = 6 + sqrt(4)
sqrt(81) = 8 + sqrt(1)
sqrt(100) = 10 + sqrt(0)

Prove that these are the only equalities existing with this pattern.

Level: Senior

Solution due to Aditya Narayan Sharma.

We are seeking t and u such that
sqrt(10t+u) = t+sqrt(u).
We have
10t+u = [t+sqrt(u)]^2 = t^2+2t*sqrt(u)+u,
10t-t^2 = 2t*sqrt(u)),
10-t = 2*sqrt(u).
t = 10-2*sqrt(u).
Thus u must be a perfect square digit, or u = 0, 1, 4, or 9.
It follows that t = 10, 8, 6, 4.
Therefore
sqrt(100) = 10 + sqrt(0),
sqrt(81) = 8 + sqrt(1),
sqrt(64) = 6 + sqrt(4),
and
sqrt(49) = 4 + sqrt(9)
are the only equalities existing with this pattern.