Integer Chord in a Polynomial Graph

Problem

Integer Chord in a Polynomial Graph, problem

Solution

Let the $x\text{-axis}$ be the horizontal axis. Let $f(x)$ be the polynomial function. Let $a$ and $b$ be the $x\text{-coordinates}$ of the two points. Since $a,$ $b,$ and the coefficients of $f(x)$ are integers, $a - b | f(a) - f(b).$

Since the distance between the 2 points is an integer, from the Pythagorean Theorem, the following is a square number:

$\displaystyle (a-b)^2+(f(a)-f(b))^2=(a-b)^2\left(1+\left(\frac{f(a)-f(b)}{a-b}\right)^2\right).$

Therefore, $\displaystyle 1+\left(\frac{f(a)-f(b)}{a-b}\right)^2$ is a square number, implying that $\displaystyle \frac{f(a)-f(b)}{a-b}=0$ so that $f(a)=f(b).$

Thus, the segment that connects the two points is parallel to the horizontal axis.

Acknowledgment

This is problem OC292 from the Canadian Crux Mathematicorum (Vol. 43(9), November 2017). Originally problem 1 from day 1 of the 2015 Spain Mathematical Olympiad. The above solution is by Steve Chow.

 

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