Problem 12 · AMC 8 Stretch
Core
Geometry & Measurement
Counting & Probability
pigeonholevisual-representation
Mark 6 points anywhere on a stick that is 1 meter long. Show that 2 of the points are less than \(\tfrac15\) of a meter (20 cm) apart.
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Answer: two points less than 1/5 m (20 cm) apart
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Hint 1 of 4
How could you cut the stick into pieces so that any two points in the same piece are automatically close?
Still stuck? Show hint 2 →
Hint 2 of 4
Cut the 1-meter stick into 5 equal pieces. How long is each piece?
Still stuck? Show hint 3 →
Hint 3 of 4
Each piece is \(\tfrac15\) meter (20 cm) long — those are your 5 boxes. You have 6 points.
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Approach: Pigeonhole — 6 points into 5 equal pieces
- Cut the 1-meter stick into 5 equal pieces. Each piece is \(\tfrac15\) meter (20 cm) long. These 5 pieces are our boxes.
- Place the 6 points and see which piece each lands in. Since \(6 > 5\), some piece holds at least 2 points.
- Two points inside one piece of length \(\tfrac15\) meter can't be more than \(\tfrac15\) meter apart.
- So two of the points are less than \(\tfrac15\) meter (20 cm) apart.
Mark:
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