Just for the record, my intuition was totally wrong. We are talking about density, not probability. The right logic is to check that

$$ \int_1^k 1/2z^2 dz = \frac{1}{2}(1 - \frac{1}{k}) $$,

and this is indeed the case.

Just for the record, my intuition was totally wrong. We are talking about density, not probability. The right logic is to check that

$$ \int_1^k f_Z(z) dz = \int_{1/k}^1 f_Z(z) = \frac{1}{2}(1 - \frac{1}{k}) $$,

and this is indeed the case.