<p>How would you even begin the following problem?</p>
<p>Triangle ABC has <bac ==“” 60=“” degrees,=“” <cba=“” <=“90” bc=“1,” and=“” ac=“”>= AB. Let H, I, and O be the orthocenter, incenter, and circumcenter of ABC respectively. Assume that the area of the pentagon BCOIH is the maximum possible. What is <CBA?</bac></p>
<p>[2011</a> AMC 12A Problems/Problem 25 - AoPSWiki](<a href=“http://www.artofproblemsolving.com/Wiki/index.php/2011_AMC_12A_Problems/Problem_25]2011”>Art of Problem Solving)</p>
<p>However the author uses some calculus to prove that the maximal value of sin x1 + sin x2 + sin x3 (where x1 + x2 + x3 = 60) occurs when x1 = x2 = x3 = 20 deg. It can be proven in an easier way using Lagrange multipliers, but I don’t see a clear non-calculus solution.</p>
<p>I wasn’t interested so much in the solution, but more about how you get it. (I’ve read the non-calculus solution, given in my AMC 12 problem series class). It’s quite complicated and involves a load of not-so-clear steps. How would you even think of doing that? What steps do you go through in your mind when you approach that problem?</p>
<p>I actually took that AMC12 – the recent AMCs have been pretty hard compared to past years. I don’t think I even looked at the last four or five problems (including this one), still qualified for AIME.</p>
<p>A lot of advanced AMC/AIME problems have not-so-intuitive solutions…I once saw a counting problem in which the fastest solution involved complex numbers and roots of unity. It takes practice…and some ingenuity.</p>
<p>What problem was that? (the counting problem)</p>
<p>It was something like, how many subsets from {1,2,3,…,1000} are there such that the sum of its elements is divisible by 5.</p>