Advent calendar 2024
20 December
p(x) is a polynomial with integer coefficients such that:
- p(0) > 0;
- if x is a real number,
4x – 9 < p(x) < x2 – 2x + 2.
What is p(23)?
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Plotting \(y = 4x - 9\) and \(y = x^2-2x+2\) gives:
As \(p(x)\) has integer coefficients, \(p(3)\) must be an integer. If you click to view a larger version of the image, you'll see that the only option for \(p(3)\) that satisfies
\(4x-9<p(x)<x^2-2x+2\) is \(p(3)=4\).
As \(p(x)>4x-9\), it must at least a polynomial of degree 1. As \(p(x)<x^2-2x+2\), it must be at most a polynomial of degree 2. Playing with coefficients, you can see that
\(p(x)\) is either \(x^2-2x+1\) or \(4x-8\). \(4x-8\) is negative when \(x=0\), so \(p(x)=x^2-2x+1\) and \(p(23)\) is 484.
![](javascript:showlimage("adv2024-ans20.png"))