Advanced book on Mathematics Olympiad

32 2 Algebra

99.Letaandbbe real numbers such that

9 a^2 + 8 ab+ 7 b^2 ≤ 6.

Prove that 7a+ 5 b+ 12 ab≤9.

100.Leta 1 ,a 2 ,...,anbe real numbers such thata 1 +a 2 +···+an≥n^2 anda 12 +a 22 +

···+an^2 ≤n^3 +1. Prove thatn− 1 ≤ak≤n+1 for allk.

101.Find all pairs(x, y)of real numbers that are solutions to the system

x^4 + 2 x^3 −y=−

1

4

+

√

3 ,

y^4 + 2 y^3 −x=−

1

4

−

√

3.

102.Letnbe an even positive integer. Prove that for any real numberxthere are at least

2 n/^2 choices of the signs+and−such that

±xn±xn−^1 ±···±x<

1

2

.

2.1.3 The Cauchy–Schwarz Inequality

A direct application of the discussion in the previous section is the proof of the Cauchy–
Schwarz (or Cauchy–Bunyakovski–Schwarz) inequality

∑n

k= 1

a^2 k

∑n

k= 1

b^2 k≥

(n

∑

k= 1

akbk

) 2

,

where the equality holds if and only if theai’s and thebi’s are proportional. The expression

∑n

k= 1

a^2 k

∑n

k= 1

b^2 k−

(n

∑

k= 1

akbk

) 2

is a quadratic function in theai’s andbi’s. For it to have only nonnegative values, it

should be a sum of squares. And this is true by the Lagrange identity

∑n

k= 1

ak^2

∑n

k= 1

bk^2 −

(n

∑

k= 1

akbk

) 2

=

∑

i<k

(aibk−akbi)^2.

Sadly, this proof works only in the finite-dimensional case, while the Cauchy–

Schwarz inequality is true in far more generality, such as for square integrable functions.

Its correct framework is that of a real or complex vector space, which could be finite or

infinite dimensional, endowed with an inner product〈·,·〉.

By definition, an inner product is subject to the following conditions: