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

Problem

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.

Solution

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 $(n,m,\overline{a},\overline{c}) = (0,4,-2,18),(1,3,-1,10),(2,6,-4,40),(3,5,-2,27),\ldots$

2017 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4

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