102 Combinatorial problems from the training of USA IMO team by Andreescu T., Feng Z.

By Andreescu T., Feng Z.

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22. t a round dinner table with men and women alternatiug but no one sits next to their spouse? You may wish to try this nontrivial problem for yourself before reading any fnrther. 2. e consider only seatings where husbauds and wives alternate. Let A, he seatings with couple i sitting together. We want to count the seatings that are in none of the A;. So, we want from the Lemma for PIE, with t = (n - 1)! n!. The G) terms in S\- are all ~~qual. \Ve suppose Adam and Eve are the first couple. \Ve compute by placing persons around the table but.

3 .... , n} for which i is in the i 111 position. \Ve need to count the arrangements that are in none of the sets Ai. By the Lemma for PIE. = n! - (n) 11h+ .. · + (-1)"~]. l ~+~-A+~1. -· ,3. ~. n.. J. Recalling· (or taking it as a definition) that e 1 ' number :r. + .. ·+(-1)" n 0! _ . r' for real ~ k 1 . + .. ·+(-I) k,+ h·=O ... \Ve see that 0 11 /n.! is very close to the number I/ e. /e for all n. So Dn:::::; n! 367879441171442 .... If letters, 1 addressed to each of some veople, are passed ont at random.

N} such that none of the following 2n conditions is satisfied? §22. Eating Out 1 is pt; 2 is 2nd; 3 is 3rd; n - 1 is (n - 1)"t; n is nth; 41 1 is 2"d; 2 is 3rd; 3 is 4th; n- 1 is nth; n is pt. Eating Out Problems. 1. Suppose n persons have both lunch and dinner together. For a given seating arrangement at lunch, verify the displayed solution to the question, How many ways are there to seat these persons for dinner such that no one has the same neighbor on the right at both meals, under the specified condition?