Problem 27 · 2012 Math Kangaroo
Stretch
Algebra & Patterns
substitution
After an especially intense lesson the graph of the function y = x² was still on the board as well as 2012 straight lines parallel to the straight line with the equation y = x, which each intersected the parabola in two points. How big is the sum of all x-coordinates of the intersections of the straight lines with the parabola?
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Answer: D — 2012
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Hint 1 of 2
Each line parallel to \(y = x\) has the form \(y = x + c\); intersect it with \(y = x^2\).
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Hint 2 of 2
The two \(x\)-values on one line are the roots of \(x^2 - x - c = 0\) — use the sum of roots, which is the same for every line.
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Approach: sum of roots per line, added over all lines
- A line \(y = x + c\) meets \(y = x^2\) where \(x^2 - x - c = 0\), whose two roots sum to \(1\) (independent of \(c\)).
- So each of the 2012 lines contributes \(1\) to the running total of \(x\)-coordinates.
- The overall sum is \(2012 \cdot 1 = 2012\), choice D.
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