Advanced book on Mathematics Olympiad

2.2 Polynomials 45

2.1.7 Other Inequalities........................................

We conclude with a section for the inequalities aficionado. Behind each problem hides a

famous inequality.

139.Ifxandyare positive real numbers, show thatxy+yx>1.

140.Prove that for alla, b, c≥0,

(a^5 −a^2 + 3 )(b^5 −b^2 + 3 )(c^5 −c^2 + 3 )≥(a+b+c)^3.

141.Assume that all the zeros of the polynomialP(x)=xn+a 1 xn−^1 +···+anare real

and positive. Show that if there exist 1≤m<p≤nsuch thatam=(− 1 )m

(n

m

)

andap=(− 1 )p

(n

p

)

, thenP(x)=(x− 1 )n.

142.Letn>2 be an integer, and letx 1 ,x 2 ,...,xnbe positive numbers with the sum

equal to 1. Prove that

∏n

i= 1

(

1 +

1

xi

)

≥

∏n

i= 1

(

n−xi

1 −xi

)

.

143.Leta 1 ,a 2 ,...,an,b 1 ,b 2 ,...,bnbe real numbers such that

(a^21 +a 22 +···+a^2 n− 1 )(b^21 +b 22 +···+b^2 n− 1 )

>(a 1 b 1 +a 2 b 2 +···+anbn− 1 )^2.

Prove thata^21 +a 22 +···+a^2 n>1 and b^21 +b^22 +···+bn^2 > 1.

144.Leta, b, c, dbe positive numbers such thatabc=1. Prove that

1

a^3 (b+c)

+

1

b^3 (c+a)

+

1

c^3 (a+b)

≥

3

2

.

2.2 Polynomials

2.2.1 A Warmup

A polynomial is a sum of the form

P(x)=anxn+an− 1 xn−^1 +···+a 0 ,

wherexis the variable, andan,an− 1 ,...,a 0 are constant coefficients. Ifan =0, the

numbernis called the degree, denoted by deg(P (x)).Ifan =1, the polynomial is

called monic. The sets, which, in fact, are rings, of polynomials with integer, rational,