Advanced book on Mathematics Olympiad

Algebra 371

a

d

b

c

a

c

b

d

a

d b c

Figure 59

Foru/∈D 2 , the triangle inequality gives

|u−λi|≥|u−c|−|c−λi|>R+|k|−R=|k|.

Hence|u−|k|λi|<1, fori= 1 , 2 ,...,n. For such auwe then have

|nP (u)−kP′(u)|=

∣

∣∣

∣∣nP (u)−kP (u)

∑n

i= 1

1

u−λi

∣

∣∣

∣∣=|P (u)|

∣

∣∣

∣∣n−k

∑n

i= 1

1

u−λi

∣

∣∣

∣∣

≥|P (u)|

∣

∣

∣∣

∣

n−

∑n

i= 1

|k|

|u−λi|

∣

∣

∣∣

∣

,

where the last inequality follows from the triangle inequality.
But we have seen that

n−

∑n

i= 1

|k|

|u−λi|

=

∑n

i= 1

(

1 −

|k|

|u−λi|

)

> 0 ,

and sinceP (u) =0, it follows thatucannot be a root ofnP (u)−kP′(u). Thus all roots
of this polynomial lie inD 2.
(17th W.L. Putnam Mathematical Competition, 1956)

116.The inequality in the statement is equivalent to

(a^2 +b^2 +c^2 )^2 < 4 (a^2 b^2 +b^2 c^2 +c^2 a^2 ).

The latter can be written as

0 <( 2 bc)^2 −(a^2 −b^2 −c^2 )^2 ,

or

( 2 bc+b^2 +c^2 −a^2 )( 2 bc−b^2 −c^2 +a^2 ).