ILL 1990 Problem 16

We call an integer $k \geq 1$ having property $P$ , if there exists at least one integer $m \geq 1$ which cannot be expressed in the form $m = \varepsilon_1 z_1^k + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k$ , where $z_i$ are nonnegative integer and $\varepsilon _i = 1$ or $-1$ , $i = 1, 2, \ldots, 2k$ . Prove that there are infinitely many integers $k$ having the property $P.$