MathDB

Let the polynomial

P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0

where

1011\leq a_i\leq 2021

for all

i=0,1,2,...,21.

Given that

P(x)

has an integer root and there exists an positive real number

c

such that

|a_{k+2}-a_k|\leq c

for all

k=0,1,...,19.

a) Prove that

P(x)

has an only integer root.
b) Prove that

\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.

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