Difference between revisions of "1975 Canadian MO Problems/Problem 6"
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== Solution == | == Solution == | ||
| + | (i) | ||
| + | Notice that for each person there will only be one configuration(achieved through rotation) for which the name card matches the guest, therefore, for the <math>15</math> possible configurations, there will be <math>15</math> correct placed cards. But since one of the configurations does not hold any correct placed card, we shall distribute <math>15</math> correct placed cards to <math>14</math> configurations, hence using the [https://artofproblemsolving.com/wiki/index.php/Pigeonhole_Principle?srsltid=AfmBOorTkqnZSbEc86ZG31ET425xJCLwSsjujr7QRLtcfy4C3L5jJKAb Pigeonhole Principle], there is at least two correct placed cards in one single configuration. | ||
| + | (ii) | ||
None yet! | None yet! | ||
| − | |||
{{Old CanadaMO box|num-b=5|num-a=7|year=1975}} | {{Old CanadaMO box|num-b=5|num-a=7|year=1975}} | ||
Latest revision as of 12:26, 30 January 2025
Problem 6
(i)
chairs are equally place around a circular table on which are name cards for quests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
chairs are equally place around a circular table on which are name cards for quests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated. (ii) Give an example of an arrangement in which just one of the 15 quests is correctly seated and for which no rotation correctly places more than one person.
Solution
(i)
Notice that for each person there will only be one configuration(achieved through rotation) for which the name card matches the guest, therefore, for the possible configurations, there will be correct placed cards. But since one of the configurations does not hold any correct placed card, we shall distribute correct placed cards to 
configurations, hence using the Pigeonhole Principle, there is at least two correct placed cards in one single configuration.
(ii)
None yet!
| 1975 Canadian MO (Problems) | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 7 |
