Trigonometry · Challenge problem

Vectors $\mathbf a$ and $\mathbf b$ have $|\mathbf a| = 1$ and $|\mathbf b| = 2$, and $|\mathbf a + \mathbf b| = |2\mathbf a - \mathbf b|$. Find the cosine of the angle between $\mathbf a$ and $\mathbf b$.