Subject : Mathematics Olympiad Assignment 3

Mathematics Olympiad

Topic - Roots & Coefficient

1. Let a, b, c be non zero real numbers such that (ab + bc + ca)3 = abc (a + b + c)3.Prove that a, b, c are terms of a geometric seqence.

2. Solve in real numbers the system of equationsx + y + z = 4x2 + y2 + z2 = 14x3 + y3 + z3 = 34

3. Let a and b be two of the roots of polynomial equation x4 + x3 – 1 = 0Prove that ab is a root of polynomial equation x6 + x3 – x2 – 1 = 0

4. Let a, b, c be non zero real numbers such that a + b + c 0 and

1 1 1 1a b c a b c

Prove that for all odd integer n

n n n n n n

1 1 1 1a b c a b c

5. Let a b c be real numbers such that a + b + c = 2 and ab + bc + ca =1

Prove that 0 a 13 b 1 c

43

6. Prove that two of the four roots of the polynomial x4 + 12x – 5 add up to 2

7. Solve in real numbers the system of equation.x + y + z = 0x3 + y3 + z3 = 18x7 + y7 + z7 = 2058

8. Solve in real numbers the system of equationa + b = 8ab + c + d = 23ad + bc = 28cd = 12