Advanced book on Mathematics Olympiad

2.2 Polynomials 51

155.Leta, b, cbe real numbers. Show thata ≥0,b ≥0, andc≥0 if and only if
a+b+c≥0,ab+bc+ca≥0, andabc≥0.

156.Solve the system

x+y+z= 1 ,

xyz= 1 ,

knowing thatx, y, zare complex numbers of absolute value equal to 1.

157.Find all real numbersrfor which there is at least one triple(x,y,z)of nonzero real
numbers such that

x^2 y+y^2 z+z^2 x=xy^2 +yz^2 +zx^2 =rxyz.

158.For five integersa, b, c, d, ewe know that the sumsa+b+c+d+eanda^2 +
b^2 +c^2 +d^2 +e^2 are divisible by an odd numbern. Prove that the expression
a^5 +b^5 +c^5 +d^5 +e^5 − 5 abcdeis also divisible byn.

159.Find all polynomials whose coefficients are equal either to 1 or−1 and whose zeros
are all real.

160.LetP(z)=az^4 +bz^3 +cz^2 +dz+e=a(z−r 1 )(z−r 2 )(z−r 3 )(z−r 4 ), where
a, b, c, d, eare integers,a =0. Show that ifr 1 +r 2 is a rational number, and if
r 1 +r 2 =r 3 +r 4 , thenr 1 r 2 is a rational number.

161.The zeros of the polynomialP(x)=x^3 − 10 x+11 areu, v, andw. Determine the
value of arctanu+arctanv+arctanw.

162.Prove that for every positive integern,

tan

π

2 n+ 1

tan

2 π

2 n+ 1

···tan

nπ

2 n+ 1

=

√

2 n+ 1.

163.LetP(x)=xn+an− 1 xn−^1 +···+a 0 be a polynomial of degreen≥3. Knowing that
an− 1 =−

(n

1

)

,an− 2 =

(n

2

)

, and that all roots are real, find the remaining coefficients.

164.Determine the maximum value ofλsuch that wheneverP(x)=x^3 +ax^2 +bx+c
is a cubic polynomial with all zeros real and nonnegative, then

P(x)≥λ(x−a)^3

for allx≥0. Find the equality condition.

165.Prove that there are unique positive integersa,nsuch that

an+^1 −(a+ 1 )n= 2001.