Problem 30 · 2015 Math Kangaroo
Stretch
Counting & Probability
careful-counting
Several points are marked on a straight line. Then all possible connecting lines between each two points are drawn. One such point lies within exactly 80 of those connecting lines, and another one lies within exactly 90 of those. How many points were marked on the straight line?
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Answer: B — 22
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Hint 1 of 2
A marked point with L points to its left and R to its right lies inside exactly L×R of the segments.
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Hint 2 of 2
Find one N giving both L×R = 80 and L×R = 90 at two points; N = 22 works (5×16 and 6×15).
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Approach: count segments through a point as L×R
- A point with L points left and R right is interior to L×R segments.
- We need L×R = 80 and (elsewhere) = 90 for the same total N.
- With N = 22: a point at position 6 gives 5×16 = 80, at position 7 gives 6×15 = 90.
- So 22 points were marked.
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