Advanced book on Mathematics Olympiad

2.1 Identities and Inequalities 27

Example.Letx, y, zbe distinct real numbers. Prove that

√ (^3) x−y+√ (^3) y−z+√ (^3) z−x = 0.
Solution.The solution is based on the identity
a^3 +b^3 +c^3 − 3 abc=(a+b+c)(a^2 +b^2 +c^2 −ab−bc−ca).
This identity comes from computing the determinant

D=

∣∣

∣∣

∣∣

abc

cab

bca

∣∣

∣∣

∣∣

in two ways: first by expanding with Sarrus’ rule, and second by adding up all columns
to the first, factoring(a+b+c), and then expanding the remaining determinant. Note
that this identity can also be written as

a^3 +b^3 +c^3 − 3 abc=

1

2

(a+b+c)

[

(a−b)^2 +(b−c)^2 +(c−a)^2

]

.

Returning to the problem, let us assume the contrary, and set^3

√

x−y=a,^3

√

y−z=
b,^3

√

z−x=c. By assumption,a+b+c=0, and soa^3 +b^3 +c^3 = 3 abc. But this
implies

0 =(x−y)+(y−z)+(z−x)= 33

√

x−y^3

√

y−z^3

√

z−x = 0 ,

since the numbers are distinct. The contradiction we have reached proves that our as-
sumption is false, and so the sum is nonzero. 

And now the problems.

81.Show that for no positive integerncan bothn+3 andn^2 + 3 n+3 be perfect cubes.

82.LetAandBbe twon×nmatrices that commute and such that for some positive

integerspandq,Ap=InandBq=On. Prove thatA+Bis invertible, and find

its inverse.

83.Prove that any polynomial with real coefficients that takes only nonnegative values

can be written as the sum of the squares of two polynomials.

84.Prove that for any nonnegative integern, the number

55

n+ 1

+ 55

n

+ 1

is not prime.