APMO 1995 Q2: Upper Secondary Mathematics Competition Question

Let a₁, a₂, . . . , a_n be a sequence of integers with values between 2 and 1995 such that:

(i) Any two of the a_i’s are relatively prime,

(ii) Each a_i is either a prime or a product of primes.

Determine the smallest possible values of n to make sure that the sequence will contain a prime number.


[APMO 1995 Q2][with minor edit of typo]

I am posting this question as I think it is interesting, however, I also feel it needs some interpretation. I have copied it as written (apart from one typo correction), but I'm not sure why it asks for "values of n" in the plural. I assume the question is asking us to firstly find the maximum number of terms such that conditions (i) and (ii) are satisfied but without any primes appearing; then by adding one unused prime to such a sequence we would have found the minimum number that guarantees that a prime be present. I also assume that the integers are 2 to 1995 inclusive. I leave these assumptions open for discussion; it may be a case of lost in translation as APMO is the Asian Pacific MO.

Feel free to comment, ask questions and even check your answer in the comments box below powered by Disqus Google+.

}

This space is here to avoid seeing the answers before trying the problem!

}

If you enjoy using this website then please consider making a donation - every little helps :-)

You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.

Don’t forget to follow Gifted Mathematics on Google+, Facebook or Twitter. You may add your own interesting questions on our Google+ Community and Facebook..

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.