# Mathematical Food for Thought

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Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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 Get A Tan. Topic: Trigonometry. Level: AMC. April 9th, 2007 Problem: (2007 MAO State – Gemini) Let be in degrees and . Solve for : . Solution: Here’s a nice tangent identity that is not very well-known, but rather cool. Start with the regular tangent angle addition identity, . Letting and , we obtain . Well, look at that. It’s the same expression as the RHS of the equation we want to solve. Substituting accordingly, it remains to solve . QED. ——————– Comment: This identity is a nice one to keep around because it can turn up unexpectedly. Especially when you see that exact form and you’re like “whoa this is such a nice form there must be an identity for it.” So there. ——————– Practice Problem: (2007 MAO State – Gemini) One hundred positive integers, not necessarily distinct, have a sum of . What is the largest possible product these numbers can attain? Posted in AMC, Trigonometry || 3 Comments » MAO Results. April 3rd, 2007 Check them out online! http://www.wamath.net/contests/StateMAT/ Posted in Announcements || 2 Comments » Sumsine. Topic: Trigonometry/Geometry. Level: AMC. April 1st, 2007 Problem: (2007 MAO State – Gemini) Find the sum of the sines of the angles of a triangle whose perimeter is five times as large as its circumradius. Solution: At first, this problem seems very strange because we do not expect a nice relationship between the perimeter and the circumradius to offer much, but take another look. Sines, circumradius… Law of Sines! In fact, we get so we can rewrite as . QED. ——————– Comment: Admittedly, MAO does not exactly have a plethora of cool problems, but this one was not bad. It wasn’t too difficult, but it required some clever use of well-known identities to pull it off. Given that half of the problems were computationally intensive (think AIME I but worse), this was a nice relief in the middle of the contest. ——————– Practice Problem: (2007 MAO State – Gemini) The zeros of form an arithmetic sequence. Let , where and are relatively prime positive integers. Find the value of . Posted in Geometry, Trigonometry || 2 Comments » 2007 MAO. March 31st, 2007 Good job to all Bellevue people at MAO State! We placed 3rd in Sweepstakes! That’s the highest we’ve ever gotten in my four years . Posted in Announcements || 1 Comment » Condensation Sensation. Topic: Calculus/S&S. March 27th, 2007 Theorem: (Cauchy Condensation Test) If is a monotonically decreasing sequence of positive reals and is a positive integer, then converges if and only if converges. ——————– Problem: Determine the convergence of , where is a positive real. Solution: Well, let’s apply the Cauchy condensation test. Then we know that converges if and only if does. But this clearly diverges due to the fact that the numerator is exponential and the denominator is a power function. QED. ——————– Comment: This is a pretty powerful test for convergence, at least in the situations in which it can be applied. The non-calculus proof for the divergence of the harmonic series is very similar to the Cauchy condensation test; in fact, the condensation test would state that converges iff converges, which clearly shows that the harmonic series diverges. ——————– Practice Problem: Determine the convergence of . Posted in Calculus, Sequences & Series || 5 Comments »