Advanced book on Mathematics Olympiad

16 1 Methods of Proof

This is equivalent to

2 x^2 − 2

√

2 x+ 1 ≥ 0 ,

or(x

√

2 − 1 )^2 ≥0, which is obvious and we are done. 

What we particularly like about the shaded square from Figure 4 is that it plays the
role of the “largest square’’ when placed on the left, and of the “smallest square’’ when
placed on the right. Here are more problems.

52.Givenn≥3 points in the plane, prove that some three of them form an angle less

than or equal toπn.

53.Consider a planar region of area 1, obtained as the union of finitely many disks.

Prove that from these disks we can select some that are mutually disjoint and have

total area at least^19.

54.Suppose thatn(r)denotes the number of points with integer coordinates on a circle

of radiusr>1. Prove that

n(r) < 2 π

√ 3

r^2.

55.Prove that among any eight positive integers less than 2004 there are four, say

a, b, c, andd, such that

4 +d≤a+b+c≤ 4 d.

56.Leta 1 ,a 2 ,...,an,...be a sequence of distinct positive integers. Prove that for

any positive integern,

a^21 +a 22 +···+a^2 n≥

2 n+ 1

3

(a 1 +a 2 +···+an).

57.LetXbe a subset of the positive integers with the property that the sum of any two

not necessarily distinct elements inXis again inX. Suppose that{a 1 ,a 2 ,...,an}

is the set of all positive integers not inX. Prove thata 1 +a 2 +···+an≤n^2.

An order on a finite set hasmaximalandminimalelements. If the order is total, the
maximal (respectively, minimal) element is unique. Quite often it is useful to look at
such extremal elements, as is the case with the following problem.

Example.Prove that it is impossible to dissect a cube into finitely many cubes, no two
of which are the same size.