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Problem 1379

AIME late

Number theory Difficulty 5.7 Prove it Final Round of National Olympiad · Estonia

Prove or disprove: For every integer $n > 0$ , there is a polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ satisfying the following conditions:
(1) All coefficients $a_n, a_{n-1}, \dots, a_1, a_0$ are positive real numbers;
(2) At least one of the coefficients is $\frac{1}{n}$ ;
(3) The value of the polynomial is integer for every integer $x$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

The product of any $n$ consecutive integers is divisible by $n$ . Thus the conditions are satisfied by the polynomial obtained by removing parentheses and collecting terms in the expression $\frac{1}{n}(x+1)\dots(x+n)$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.