103 Trigonometry Problems

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3. Advanced Problems 77 22. Let a 0 = √ 2 + √ 3 + √ 6, and let a n+1 = a2 n −5 2(a n +2) for integers n>0. Prove that ( 2 n−3 ) π a n = cot − 2 3 for all n. 23. Let n be an integer with n ≥ 2. Prove that n∏ k=1 [ )] π tan (1 + 3k 3 3 n = − 1 n∏ k=1 [ )] π cot (1 − 3k 3 3 n . − 1 24. Let P 2 (x) = x 2 − 2. Find all sequences of polynomials {P k (x)} ∞ k=1 such that P k (x) is monic (that is, with leading coefficient 1), has degree k, and P i (P j (x)) = P j (P i (x)) for all i and j. 25. In triangle ABC, a ≤ b ≤ c.As a function of angle C, determine the conditions under which a + b − 2R − 2r is positive, negative, or zero. 26. Let ABC be a triangle. Points D, E, F are on sides BC,CA,AB, respectively, such that |DC|+|CE|=|EA|+|AF |=|FB|+|BD|. Prove that |DE|+|EF|+|FD|≥ 1 2 (|AB|+|BC|+|CA|). 27. Let a and b be positive real numbers. Prove that 1 √ + 1 √ ≥ 2 √ 1 + a 2 1 + b 2 1 + ab if either (1) 0 |AC| and ̸ B = 45 ◦ . Let O and I denote the circumcenter and incenter of triangle ABC, respectively. Suppose that √ 2|OI| =|AB| −|AC|. Determine all the possible values of sin A.
78 103 Trigonometry Problems 29. Let n be a positive integer. Find the real numbers a 0 and a k,l ,1≤ l
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About the Authors Titu Andreescu re
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Titu Andreescu University of Wiscon
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vi Contents Vectors 41 The Dot Prod
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viii Preface Throughout MOSP, full
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Abbreviations and Notation Abbrevia
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103 Trigonometry Problems
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2 103 Trigonometry Problems First w
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4 103 Trigonometry Problems triangl
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5. Solutions to Advanced Problems 1
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or 5. Solutions to Advanced Problem
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It follows that ∑ 4 sin 3 A cos(B
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200 103 Trigonometry Problems Binom
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Further Reading 1. Andreescu, T.; F
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Further Reading 213 23. Fomin, D.;
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