HMMT February

Algebra Difficulty 5.1 AIME, harder Find the answer

Let $a \star b=\sin a \cos b$ for all real numbers $a$ and $b$ . If $x$ and $y$ are real numbers such that $x \star y-y \star x=1$ , what is the maximum value of $x \star y+y \star x$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have $x \star y+y \star x=\sin x \cos y+\cos x \sin y=\sin (x+y) \leq 1$ . Equality is achieved when $x=\frac{\pi}{2}$ and $y=0$ . Indeed, for these values of $x$ and $y$ , we have $x \star y-y \star x=\sin x \cos y-\cos x \sin y=\sin (x-y)=\sin \frac{\pi}{2}=1$ .

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