MathDB

1995 May Olympiad

Part of May Olympiad

Subcontests

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palindromic numbers

It is initially considered a number of three different digits, none of which is equal to zero. Changing instead two of its digits meet a second number less than the first. If the difference between the first and second is a two-digit number and the sum of the first and the second is a palindromic number less than 500500, what are the palindromics that can be obtained?

Rodolfo and Gabriela play with 9 numbered chips

Rodolfo and Gabriela have 99 chips numbered from 11 to 99 and they have fun with the following game: They remove the chips one by one and alternately (until they have 33 chips each), with the following rules: ∙\bullet Rodolfo begins the game, choosing a chip and in the following moves he must remove, each time, a chip three units greater than the last chip drawn by Gabriela. ∙\bullet Gabriela, on her turn, chooses a first chip and in the following times she must draw, each time, a chip two units smaller than the last chip that she herself drew. ∙\bullet The game is won by whoever gets the highest number by adding up their three tokens. ∙\bullet If the game cannot be completed, a tie is declared. If they play without making mistakes, how should Rodolfo play to be sure he doesn't lose?
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many screws in closed boxes

The owner of a hardware store bought a quantity of screws in closed boxes and sells the screws separately. He never has more than one open box. At the end of the day Monday there are 22082208 screws left, at the end of the day Tuesday there are still 16161616 screws and at the end of Wednesday there are still 973973 screws. To control the employees, every night he writes down the number of screws that are in the only open box. The amount entered on Tuesday is double that of Monday. How many screws are there in each closed box if it is known that they are less than 500500?

289 coins stored in boxes from different countries

Julia has 289289 coins stored in boxes: All the boxes contain the same number of coins (which is greater than 11) and in each box there are coins from the same country, The coins from Bolivia are more than 6%6\% of the total, those from Chile are more than 12%12\% of the total, those of Mexico are more than 24%24\% of the total and those of Peru more than 36%36\% of the total. Can Julia have any coins from Uruguay?
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2 player game with largest prime divisors

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud 11 wins , at which point the game ends. Ana chose a number greater than 5050 and less than 100100 and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

admissions in a secret society

The management of a secret society is made up of 44 people. To admit new partners they use the following criteria: ∙\bullet Only the 44 members of the directory vote, being able to do it in 33 ways: in favor, against or abstaining. ∙\bullet Each aspiring partner must obtain at least 22 votes in favor and none against. At the last management meeting, 88 requests for admission were examined. Of the total votes cast, there were 2323 votes in favor, 22 votes against and 77 abstaining. What is the highest and what is the lowest value that the number of approved admission requests can have on that occasion?
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