Page 291 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Examples 277
(c) Suppose that n 0 ≥ 2 and the arithmetic-geometric mean inequality
fails for some list of length n 0 . In other words, there are positive real
such that
numbers a 1 , a 2 , . .., a n 0
√
a 1 + a 2 +· · · + a n 0
< n 0 a 1 a 2 ··· a n 0 .
n 0
Prove that for all n ≥ n 0 , the arithmetic-geometric mean inequality
fails for some list of length n.
(d) Prove that the arithmetic-geometric mean inequality always holds.
9. Prove that if n ≥ 2 and a 1 , a 2 ,. . . , a n is a list of positive real numbers,
then
n √
≤ n a 1 a 2 ··· a n .
1 1 1
+ +· · · +
a 1 a 2 a n
(Hint: Apply exercise 8. The number on the left side of the inequality
above is called the harmonic mean of the numbers a 1 , a 2 , . .., a n .)
∗ n
10. Prove that for every set A,if A has n elements then P (A) has 2 elements.
11. If A is a set, let P 2 (A) be the set of all subsets of A that have exactly two
elements. Prove that for every set A,if A has n elements then P 2 (A) has
n(n − 1)/2 elements. (Hint: See the solution for exercise 10.)
12. Suppose n is a positive integer. An equilateral triangle is cut into 4 n
congruent equilateral triangles, and one corner is removed. (Figure 10
shows an example in the case n = 2.) Show that the remaining area can
be covered by trapezoidal tiles like this: .

Figure 10
13. Let n be a positive integer. Suppose n chords are drawn in a circle in such
∗
a way that each chord intersects every other, but no three intersect at one
2
point. Prove that the chords cut the circle into n +n+2 regions. (Figure 11
2 2
shows an example in the case n = 4. Note that there are 4 +4+2 = 11
2
regions in this figure.)
14. Let n be a positive integer, and suppose that n chords are drawn in a
circle, cutting the circle into a number a regions. Prove that the regions
can be colored with two colors in such a way that adjacent regions (that