A mass $m$ is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

In a vertical circular motion, the tension ($T$) in the wire at any point is given by:
$$T=\frac{m{v}^2}{r}+mg\cos \theta$$
where $\theta$ is the angle made with the downward vertical.At the lowest point ($\theta =0^{\circ }$): $T_{low}=\frac{m{v}^2}{r}+mg$.At the highest point ($\theta =180^{\circ }$): $T_{high}=\frac{m{v}^2}{r}-mg$.Since the tension is maximum at the lowest point, the wire is most likely to break there.