1935 Moscow University Mathematical Olympiad, Problem 6

{:description a system in three variables with a forced single solution:)

How many real solution does the following system have?

\Big\{\array{x+y=2 \\ xy-z^2=1}

Solution

The key to the solution is the Arithmetic Mean - Geometric Mean Inequality:

\frac{x+y}{2} \ge \sqrt{xy}.

Indeed, since x + y = 2, xy \le 1. On the other hand, from the second inequality xy = 1 + z^2 \ge 1, which means that xy = 1. Together with x + y = 2, this gives x = y = 1, implying z = 0.