1977 AHSME Problems/Problem 29
Find the smallest integer such that for all real numbers , and .
We see squares and one number. And we see an inequality. This calls for Cauchy's inequality. EEEEWWW.
Anyways, look at which side is which. The squared side is smaller-- so that's good. It's in the right format.
Cauchy's states that
Therefore, we see that, if we equate we get the equality right away. What's the final step? Figuring out this n. Now, note that the equation is basically complete; all we need is for . So each of them is just 1, and -- answer choice !