<p>The Art of Problem Solving (AOPS) website mentioned by tokenadult is a terrific resource.</p>
<p>AOPS also administers the USAMTS (USA Mathematics Talent Search), a proof-based competition offered four times a year. It’s an individual competition which any student in the country can enter for free. It doesn’t require the school to register or any bureaucratic hurdles. There’s no travel involved. And unlike most other math contests, there’s no time pressure. Each round of the competition gives students a month to work on and submit solutions to 5 problems. </p>
<p>Details are at <a href=“http://www.usamts.org/[/url]”>http://www.usamts.org/</a></p>
<p>Also, ARML (<a href=“http://www.arml.com/)%5B/url%5D”>http://www.arml.com/)</a>, now offers a “Challenge of the Month,” a proof-based challenge that individuals can participate in for free (no fees or bureaucratic hassles; no team needed). In addition, ARML offers school math teams the option of participating in the proof-based ARML Power Contest by mail in Nov & Feb. And of course, 1500 students a year travel in regional teams to ARML’s three national competition sites in early June, where the Power Round requires proofs. </p>
<p>Also, some colleges have Putnam preparation clubs for their undergrads and some of these clubs welcome strong local high school students to partiicipate as well.</p>