Advanced book on Mathematics Olympiad

2.1 Identities and Inequalities 31

when expanded gives rise to the following terms:

aij^2 +akl^2 +ail^2 +a^2 kj+ 2 aijakl+ 2 ailakj− 2 ailaij− 2 aijakj− 2 aklail− 2 aklakj.

For a fixed pair(i, j ), the termaijappears in(n− 1 )^2 such expressions. The products
2 aijakland 2ailakjappear just once, while the products 2ailaij,2aijakj,2aklail,2aklakj
appear(n− 1 )times (once for each square of the form(i, j ), (i, l), (k, j ), (k, l)). It
follows that the expression that we are trying to prove is nonnegative is nothing but
∑

ij kl

(aij+akl−ail−akj)^2 ,

which is of course nonnegative. This proves the inequality for all Riemann sums of the
functionf, and hence forfitself. 

94.Find

min

a,b∈R

max(a^2 +b, b^2 +a).

95.Prove that for all real numbersx,

2 x+ 3 x− 4 x+ 6 x− 9 x≤ 1.

96.Find all positive integersnfor which the equation

nx^4 + 4 x+ 3 = 0

has a real root.

97.Find all triples(x,y,z)of real numbers that are solutions to the system of equations

4 x^2

4 x^2 + 1

=y,

4 y^2

4 y^2 + 1

=z,

4 z^2

4 z^2 + 1

=x.

98.Find the minimum of

logx 1

(

x 2 −

1

4

)

+logx 2

(

x 3 −

1

4

)

+···+logxn

(

x 1 −

1

4

)

,

over allx 1 ,x 2 ,...,xn∈(^14 , 1 ).