Advanced book on Mathematics Olympiad

38 2 Algebra

If Rezi≤0, then| 3 −zi|≥3. On the other hand, if|zi|<2, then by the triangle
inequality| 3 −zi|≥ 3 −|zi|>1. Hence|Q( 3 )|is a product of terms greater than 1,
and the conclusion follows. 

More applications follow.

113. Leta, b, cbe the side lengths of a triangle with the property that for any positive
     integern, the numbersan,bn,cncan also be the side lengths of a triangle. Prove
     that the triangle is necessarily isosceles.


114. Given the vectorsa, b,cin the plane, show that

‖a‖+‖b‖+‖c‖+‖a+b+c‖≥‖a+b‖+‖a+c‖+‖b+c‖.

115. LetP(z)be a polynomial with real coefficients whose roots can be covered by a
     disk of radiusR. Prove that for any real numberk, the roots of the polynomial
     nP (z)−kP′(z)can be covered by a disk of radiusR+|k|, wherenis the degree
     ofP(z), andP′(z)is the derivative.


116. Prove that the positive real numbersa, b, care the side lengths of a triangle if and
     only if

a^2 +b^2 +c^2 < 2

√

a^2 b^2 +b^2 c^2 +c^2 a^2.

117. LetABCDbe a convex cyclic quadrilateral. Prove that

|AB−CD|+|AD−BC|≥ 2 |AC−BD|.

118. LetV 1 ,V 2 ,...,VmandW 1 ,W 2 ,...,Wmbe isometries ofRn(m, npositive inte-
     gers). Assume that for allxwith‖x‖≤1,‖Vix−Wix‖≤1,i = 1 , 2 ,...,m.
     Prove that
     ∥∥
     ∥∥
     ∥

(m

∏

i= 1

Vi

)

x−

(m

∏

i= 1

Wi

)

x

∥∥

∥∥

∥

≤m,

for allxwith‖x‖≤1.

119. Given an equilateral triangleABCand a pointPthat does not lie on the circumcircle
     ofABC, show that one can construct a triangle with sides the segmentsPA,PB,
     andPC.IfPlies on the circumcircle, show that one of these segments is equal to
     the sum of the other two.

120.LetMbe a point in the plane of the triangleABCwhose centroid isG. Prove that

MA^3 ·BC+MB^3 ·AC+MC^3 ·AB≥ 3 MG·AB·BC·CA.