Page 290 - HOW TO PROVE IT: A Structured Approach, Second Edition

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276 Mathematical Induction
(b) Consider a tournament in which each contestant plays every other
contestant exactly once, and one of them wins. We’ll say that a con-
testant x is excellent if, for every other contestant y, either x beats y
or there is a third contestant z such that x beats z and z beats y. Prove
that there is at least one excellent contestant.
n
5. For each n ∈ N, let F n = 2 (2 ) + 1. (These numbers are called the Fermat
numbers, after the French mathematician Pierre de Fermat (1601–1665).
Fermat showed that F 0 , F 1 , F 2 , F 3 , and F 4 are prime, and conjectured
that all of the Fermat numbers are prime. However, over 100 years later
Euler showed that F 5 is not prime. It is not known if there is any n > 4
for which F n is prime.)
Prove that for all n ≥ 1, F n = (F 0 · F 1 · F 2 ··· F n−1 ) + 2.
6. Prove that if n ≥ 1 and a 1 , a 2 ,..., a n are any real numbers, then |a 1 +
a 2 +· · · + a n |≤|a 1 |+|a 2 |+· · · +|a n |. (Note that this generalizes the
triangle inequality; see exercise 12(c) of Section 3.5.)
7. (a) Prove that if a and b are positive real numbers, then a/b + b/a ≥ 2.
2
(Hint: Start with the fact that (a − b) ≥ 0.)
(b) Suppose that a, b, and c are real numbers and 0 < a ≤ b ≤ c.
Prove that b/c + c/a − b/a ≥ 1. (Hint: Start with the fact that
(c − a)(c − b) ≥ 0.)
(c) Prove that if n ≥ 2 and a 1 , a 2 , ..., a n are real numbers such that
0 < a 1 ≤ a 2 ≤ ... ≤ a n , then a 1 /a 2 + a 2 /a 3 +· · · + a n−1 /a n +
a n /a 1 ≥ n.
8. If n ≥ 2 and a 1 , a 2 ,. . . , a n is a list of positive real numbers, then the
∗
number (a 1 + a 2 +· · · + a n )/n is called the arithmetic mean of the num-
√
bers a 1 , a 2 ,. . . , a n , and the number n a 1 a 2 ··· a n is called their geometric
mean. In this exercise you will prove the arithmetic-geometric mean in-
equality, which says that the arithmetic mean is always at least as large
as the geometric mean.
(a) Prove that the arithmetic-geometric mean inequality holds for lists of
numbers of length 2. In other words, prove that for all positive real
√
numbers a and b,(a + b)/2 ≥ ab.
(b) Prove that the arithmetic-geometric mean inequality holds for any
list of numbers whose length is a power of 2. In other words, prove
that for all n ≥ 1, if a 1 , a 2 ,. . . , a 2 is a list of positive real numbers,
n
then

a 1 + a 2 +· · · + a 2 n 2 n √
≥ a 1 a 2 ··· a 2 .
n
2 n