2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4


Find a second-degree polynomial with integer coefficients, $p(x) = ax^2 + bx + c$, such that $p(1),p(3),p(5)$, and $p(7)$ are perfect squares, but $p(2)$ is not.


Answer: A polynomial that satisfies the criteria is easily constructed by first centering it at $x = 4$, that is $p(x) = \overline{a}(x-4)^2 + \overline{c}$. Now we have two conditions: $n^2 = p(1) = 9\overline{a}+\overline{c}$ and $m^2 = p(3) = \overline{a}+\overline{c}$ that determines possible candidates that can then be checked against the condition that $p(2)$ is not a perfect square and the condition that the coefficients are integers. A few possible answers are $(n,m,\overline{a},\overline{c}) = (0,4,-2,18),(1,3,-1,10),(2,6,-4,40),(3,5,-2,27),\ldots$

See also

2017 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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