103 Trigonometry Problems

Glossary 205
Periodic Function
A function f(x) is periodic with period T > 0ifT is the smallest positive real
number for which
f(x+ T)= f(x)
for all x.
Pigeonhole Principle
If n objects are distributed among k < n boxes, some box contains at least two
objects.
Power Mean Inequality
Let a 1 ,a 2 ,...,a n be any positive numbers for which a 1 + a 2 +···+a n = 1. For
positive numbers x 1 ,x 2 ,...,x n we define
M −∞ = min{x 1 ,x 2 ,...,x k },
M ∞ = max{x 1 ,x 2 ,...,x k },
M 0 = x a 1
1 xa 2
2 ···xa n
n ,
where t is a nonzero real number. Then
for s ≤ t.
M t = ( a 1 x t 1 + a 2x t 2 +···+a kx t k) 1/t ,
M −∞ ≤ M s ≤ M t ≤ M ∞
Rearrangement Inequality
Let a 1 ≤ a 2 ≤···≤a n ; b 1 ≤ b 2 ≤···≤b n be real numbers, and let c 1 ,c 2 ,...,c n
be any permutations of b 1 ≤ b 2 ≤···≤b n . Then
a 1 b n + a 2 b n−1 +···+a n b 1 ≤ a 1 c 1 + a 2 c 2 +···+a n c n
≤ a 1 b 1 + a 2 b 2 +···+a n b n ,
with equality if and only if a 1 = a 2 =···=a n or b 1 = b 2 =···=b n .
Root Mean Square–Arithmetic Mean Inequality
For positive numbers x 1 ,x 2 ,...,x n ,
√
x 2 1 + x2 2 +···+x2 k
n
≥ x 1 + x 2 +···+x k
.
n