Difference between revisions of "2016 IMO Problems/Problem 6"
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There are <math>n\ge 2</math> line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands <math>n-1</math> times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. | There are <math>n\ge 2</math> line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands <math>n-1</math> times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. | ||
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(b) Prove that Geoff can never fulfill his wish if <math>n</math> is even. | (b) Prove that Geoff can never fulfill his wish if <math>n</math> is even. | ||
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| + | ==Solution== | ||
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| + | ==See Also== | ||
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| + | {{IMO box|year=2016|num-b=5|after=Last Problem}} | ||
Latest revision as of 00:37, 19 November 2023
Problem
There are 
line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands 
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if 
is odd.
(b) Prove that Geoff can never fulfill his wish if is even.
Solution
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See Also
| 2016 IMO (Problems) • Resources | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Problem |
| All IMO Problems and Solutions | ||
