HMMT-Nov 2012 gen, Problem 10

Let $\alpha$ and $\beta$ be reals. Find the least possible value of \[ (2 \cos \alpha+5 \sin \beta-8)^{2}+(2 \sin \alpha+5 \cos \beta-15)^{2} \]

Solution: Let the vector $\vec{v}=(2 \cos \alpha, 2 \sin \alpha)$ and $\vec{w}=(5 \sin \beta, 5 \cos \beta)$. The locus of ends of vectors expressible in the form $\vec{v}+\vec{w}$ are the points which are five units away from a point on the circle of radius two about the origin. The expression that we desire to minimize is the square of the distance from this point to $X=(8,15)$. Thus, the closest distance from such a point to $X$ is when the point is 7 units away from the origin along the segment from the origin to $X$. Thus, since $X$ is 17 units away from the origin, the minimum is $10^{2}=100$. $\fbox{100}$.

Answer: 100