1

Part of 1995 May Olympiad

Problems(2)

2 player game with largest prime divisors

Source: I May Olympiad (Olimpiada de Mayo) 1995 L2 P1

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

1

wins , at which point the game ends. Ana chose a number greater than

50

100

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.

number theorygamegame strategyDivisors

admissions in a secret society

Source: I May Olympiad (Olimpiada de Mayo) 1995 L1 P1

The management of a secret society is made up of

4

people. To admit new partners they use the following criteria:

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4

members of the directory vote, being able to do it in

3

ways: in favor, against or abstaining.

\bullet

Each aspiring partner must obtain at least

2

votes in favor and none against. At the last management meeting,

8

requests for admission were examined. Of the total votes cast, there were

23

votes in favor,

2

votes against and

7

abstaining. What is the highest and what is the lowest value that the number of approved admission requests can have on that occasion?