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Polynomial permutation

Source: Iran TST1 Day1 P3

February 22, 2020

algebrapolynomialIranian TST

Problem Statement

We call a number

n

interesting if for each permutation

\sigma

of

1,2,\ldots,n

there exist polynomials

P_1,P_2,\ldots ,P_n

and

\epsilon > 0

such that:

i)

P_1(0)=P_2(0)=\ldots =P_n(0)

ii)

P_1(x)>P_2(x)>\ldots >P_n(x)

for

-\epsilon<x<0

iii)

P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x)

for

0<x<\epsilon

Find all interesting

n

.
Proposed by Mojtaba Zare Bidaki

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