Problem 23 · 2016 Math Kangaroo
Stretch
Counting & Probability
careful-counting
How many quadratic functions \(y = ax^2 + bx + c\) (with \(a \ne 0\)) have graphs that go through at least 3 of the marked points?
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Answer: D — 22
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Hint 1 of 2
A parabola needs three points with different x-values, so pick one point from each column.
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Hint 2 of 2
Count those 27 triples, then drop the collinear ones (which would force a = 0).
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Approach: one point per column, then subtract the lines
- For \(y = ax^2+bx+c\) the three points need distinct \(x\), so choose one point from each of the 3 columns: \(3 \times 3 \times 3 = 27\) triples.
- A triple fails to give a genuine quadratic only when the three chosen points are collinear (then \(a = 0\) or no parabola fits).
- Counting the collinear column-triples among the marked points and removing them leaves 22 graphs (option D).
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