Competition problem — Algebra — Maths Olympiad Prep

Library / /348 of 860

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. Spacing and $ signs are ignored.

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 $x \star y-y \star x=\sin x \cos y-\cos x \sin y=\sin (x-y)=\sin \frac{\pi}{2}=1$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.