103 Trigonometry Problems

3. Advanced Problems 81
47. Let n be a fixed positive integer. Determine the smallest positive real number
λ such that for any θ 1 ,θ 2 ,...,θ n in the interval ( 0, π 2
)
,if
tan θ 1 tan θ 2 ···tan θ n = 2 n/2 ,
then
cos θ 1 + cos θ 2 +···+cos θ n ≤ λ.
48. Let ABC be an acute triangle. Prove that
(sin 2B + sin 2C) 2 sin A + (sin 2C + sin 2A) 2 sin B
+ (sin 2A + sin 2B) 2 sin C ≤ 12 sin A sin B sin C.
49. On the sides of a nonobtuse triangle ABC are constructed externally a square
P 4 , a regular m-sided polygon P m , and a regular n-sided polygon P n . The
centers of the square and the two polygons form an equilateral triangle. Prove
that m = n = 6, and find the angles of triangle ABC.
50. Let ABC be an acute triangle. Prove that
( ) cos A 2
+
cos B
( ) cos B 2
+
cos C
( ) cos C 2
+ 8 cos A cos B cos C ≥ 4.
cos A
51. For any real number x and any positive integer n, prove that
n∑ sin kx
∣ k ∣ ≤ 2√ π.
k=1