AIME I 2010 #6

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] be a quadratic polynomial with real coefficients satisfying [unparseable or potentially dangerous latex formula] for all real numbers [unparseable or potentially dangerous latex formula], and suppose [unparseable or potentially dangerous latex formula]. Find [unparseable or potentially dangerous latex formula].

[klebian]

We note that the inequality can be written as [unparseable or potentially dangerous latex formula], or [unparseable or potentially dangerous latex formula]. We see, then, that if we have a function Q such that [unparseable or potentially dangerous latex formula], Q must be of the form [unparseable or potentially dangerous latex formula] (I think there's some calculus theorem that definitively states this, but its somewhat intuitive and easy to see if you graph it). We know that [unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

[stupidityismygam]