Difference between revisions of "2017 IMO Problems/Problem 5"
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An integer <math>N \ge 2</math> is given. A collection of <math>N(N + 1)</math> soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove <math>N(N - 1)</math> players from this row leaving a new row of <math>2N</math> players in which the following <math>N</math> conditions hold: | An integer <math>N \ge 2</math> is given. A collection of <math>N(N + 1)</math> soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove <math>N(N - 1)</math> players from this row leaving a new row of <math>2N</math> players in which the following <math>N</math> conditions hold: | ||
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(<math>1</math>) no one stands between the two tallest players, | (<math>1</math>) no one stands between the two tallest players, | ||
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(<math>2</math>) no one stands between the third and fourth tallest players, | (<math>2</math>) no one stands between the third and fourth tallest players, | ||
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<math>\;\;\vdots</math> | <math>\;\;\vdots</math> | ||
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(<math>N</math>) no one stands between the two shortest players. | (<math>N</math>) no one stands between the two shortest players. | ||
Show that this is always possible. | Show that this is always possible. | ||
Revision as of 06:10, 17 December 2017
An integer 
is given. A collection of 
soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove 
players from this row leaving a new row of 
players in which the following 
conditions hold:
(
) no one stands between the two tallest players,
(
) no one stands between the third and fourth tallest players,
() no one stands between the two shortest players.
Show that this is always possible.
