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Expected value with polynomials

Source: ICMC 7 Round 2 Problem 3

March 12, 2024

polynomialreal analysisexpected value

Problem Statement

Let

N{}

be a fixed positive integer,

S{}

be the set

\{1, 2,\ldots , N\}

and

\mathcal{F}

be the set of functions

f:S\to S

such that

f(i)\geqslant i

for all

i\in S.

For each

f\in\mathcal{F}

let

P_f

be the unique polynomial of degree less than

N{}

satisfying

P_f(i) = f(i)

for all

i\in S.

If

f{}

is chosen uniformly at random from

\mathcal{F}

determine the expected value of

P_f'(0)

where

P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.

Proposed by Ishan Nath

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