103 Trigonometry Problems

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5. Solutions to Advanced Problems 155 It follows that ( sin 2 ) 2 ( ) nx sin nx sin(n + 1)x 2 ( sin nx cos(n + 1)x sin 2 = + x sin x sin x On the other hand, = s 2 + c 2 . s 2 + c 2 = (sin 2x + sin 4x +···+sin 2nx) 2 + (cos 2x + cos 4x +···+cos 2nx) 2 = n + ∑ (2 sin 2lx sin 2kx + 2 cos 2lx cos 2kx) = n + 2 1≤l
156 103 Trigonometry Problems Note that |AP |= x r , |BQ|= y r , and |CR| = r z . Then the equation given in the problem statement becomes 2x + 5y + 5z = 6. (∗∗) Eliminating x from equations (∗) and (∗∗) yields Completing the squares, we obtain 5y 2 + 5z 2 + 8yz − 6y − 6z + 2 = 0. (3y − 1) 2 + (3z − 1) 2 = 4(y − z) 2 . Setting 3y − 1 = u and 3z − 1 = v gives y = u+1 3 and z = v+1 3 , and so y − z = u−v 3 . The above equation becomes 5u 2 + 8uv + 5v 2 = 0. Because the discriminant of this quadratic equation is 8 2 −4·25 < 0, the only real solution to the equation is u = v = 0. Thus there is only one possible set of values for the tangents of half-angles of ABC (namely, x = 4 3 and y = z = 1 3 ). Thus all triangles in S are isosceles and similar to one another. Indeed, we have x = |AP r | = 4 3 and y = z = |BQ| r = |CQ| r = 3 1 = 12 4 ,sowe can set r = 4, |AP |=|AR| =3, and |BP|=|BQ|=|CQ|=|CR|=12. This leads to |AB| =|AC| =15 and |BC|=24. By scaling, all triangles in S are similar to the triangle with side lengths 5, 5, 8. We can also use the half-angle formulas to calculate sin B = sin C = 2 tan C 2 1 + tan 2 C 2 = 3 5 . From this it follows that |AQ| :|QB| :|BA|=3 : 4 : 5 and |AB| :|AC| : |BC|=5 : 5 : 8. 31. [TST 2003] Let a,b,c be real numbers in the interval (0, π 2 ). Prove that sin a sin(a − b) sin(a − c) sin b sin(b − c) sin(b − a) + sin(b + c) sin(c + a) sin c sin(c − a)sin(c − b) + ≥ 0. sin(a + b)
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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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3 Advanced Problems 1. Two exercise
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3. Advanced Problems 75 9. Find the
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3. Advanced Problems 77 22. Let a 0
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Further Reading 1. Andreescu, T.; F
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Further Reading 213 23. Fomin, D.;
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103 Trigonometry Problems

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