1990 AHSME Problems/Problem 29
Contents
Problem
A subset of the integers has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
Solution 1
Notice that inclusion of the integers from to is allowed as long as no integer between and inclusive is within the set. This provides a total of = 67 solutions.
Further analyzation of the remaining integers between and , we notice that we can include all the numbers except (as including would force us to remove both and ) to obtain the maximum number of solutions.
Thus, , yielding our answer,
Solution 2
Write down in a column the elements which are indivisible by three, and then follow each one by
We can take at most elements from the first row, and at most elements from each of the next seven rows. After that we can take only from any following row. Thus the answer is the number of integers between and inclusive which are indivisible by three.
There are multiples of three in that range, so there are non-multiples, and , which is
See also
1990 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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