10.4

Part of 2011 All-Russian Olympiad Regional Round

Problems(1)

Problem 4 of Russian Regional Olympiad 2011, grade 10

Source: Problem 4 of Russian Regional Olympiad 2011, grade 10 (day 1

Non-zero real numbers

a

b

c

are such that any two of the three equations

ax^{11}+bx^4+c=0

bx^{11}+cx^4+a=0

cx^{11}+ax^4+b=0

have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)

algebra proposedalgebra