5

Part of 1989 IMO Longlists

Problems(2)

Source: IMO Longlist 1989, Problem 5

a_0, a_1, \ldots

b_0, b_1, \ldots

are defined for n \equal{} 0, 1, 2, \ldots by the equalities a_0 \equal{} \frac {\sqrt {2}}{2}, a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} and b_0 \equal{} 1, b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} Prove the inequalities for every n \equal{} 0, 1, 2, \ldots 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.

inequalitiesinductiontrigonometrylimitLaTeXalgebra unsolvedalgebra

Prove two properties for functions

Source: IMO Longlist 1989, Problem 104

n > 1

be a fixed integer. Define functions f_0(x) \equal{} 0, f_1(x) \equal{} 1 \minus{} \cos(x), and for

k > 0,

f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x). If F(x) \equal{} \sum^n_{r\equal{}1} f_r(x), prove that (a)

0 < F(x) < 1

for 0 < x < \frac{\pi}{n\plus{}1}, and (b)

F(x) > 1

for \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.

functionalgebra unsolvedalgebra