MathDB

2017 May Olympiad

Part of May Olympiad

Subcontests

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shortest words in 2 letter language (Olimpiada de Mayo) 2017 L2 P5

Ababa plays with a word made up of the letters of his name and has set certain rules: If you find an AA followed immediately by a BB, you can substitute BAABAA for them. If you find two consecutive BB's, you can delete them. If you find three consecutive AA's, you can delete them. Ababa begins with the word ABABABAABAABABABABAABAAB. With the above rules, how many letters do you have the shortest word you can come up with? Why can't you come up with one more word shorter?

24 pos. integers wanted, a+b divides ab 2017 May Olympiad L1 p5

We will say that two positive integers aa and bb form a suitable pair if a+ba+b divides abab (its sum divides its multiplication). Find 2424 positive integers that can be distribute into 1212 suitable pairs, and so that each integer number appears in only one pair and the largest of the 2424 numbers is as small as possible.
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How many of them are divisible by $7$?

We consider all 77-digit numbers that are obtained by swapping in all ways Possible digits of 12345671234567. How many of them are divisible by 77?

2 player game with red, bluec chips on n-gon 2017 May Olympiad L1 p4

Let nn be an even integer greater than 22. On the vertices of a regular polygon with n sides we can place red or blue chips. Two players, AA and BB, play alternating turns of the next mode: each player, on their turn, chooses two vertices that have no tiles and places on one of them a red chip and in the other a blue chip. The goal of AA is to get three vertices consecutive with tiles of the same color. BB's goal is to prevent this from happening. To the beginning of the game there are no tiles in any of the vertices. Show that regardless of who starts to play, Player BB can always achieve his goal.
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