HMMT Spring, 2021

Number theory Difficulty 5.1 AIME, harder Prove it United States

Problem:

Among all polynomials $P(x)$ with integer coefficients for which $P(-10)=145$ and $P(9)=164$ , compute the smallest possible value of $|P(0)|$ .

Solution

Solution:

Since $a-b \mid P(a)-P(b)$ for any integer polynomial $P$ and integers $a$ and $b$ , we require that $10 \mid P(0)-P(-10)$ and $9 \mid P(0)-P(9)$ . So, we are looking for an integer $a$ near $0$ for which
$a \equiv 5 \bmod 10, \quad a \equiv 2 \bmod 9$
The smallest such positive integer is $65$ , and the smallest such negative integer is $-25$ . This is achievable, for example, if $P(x)=2x^{2}+3x-25$ , so our answer is $25$ .

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