Difference between revisions of "1986 AIME Problems/Problem 12"
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The maximum is <math>\boxed{061}</math>, attained when <math>S=\{ 15,14,13,11,8\}</math>. We must now prove that no such set has sum at least 62. Suppose such a set <math>S</math> existed. Then <math>S</math> must have more than 4 elements, otherwise its sum would be at most <math>15+14+13+12=54</math> if it had 4 elements. | The maximum is <math>\boxed{061}</math>, attained when <math>S=\{ 15,14,13,11,8\}</math>. We must now prove that no such set has sum at least 62. Suppose such a set <math>S</math> existed. Then <math>S</math> must have more than 4 elements, otherwise its sum would be at most <math>15+14+13+12=54</math> if it had 4 elements. | ||
| − | But also, <math>S</math> can't have at least 6 elements. To see why, note that <math>1 + 6 + \dbinom{6}{2} + \dbinom{6}{3} + \dbinom{6}{4}=57</math> of its subsets have at most four elements (the number of subsets with no elements plus the number of subsets with one element plus the number of subsets with two elements plus the number of subsets with three elements plus the number of subsets with four elements), so each of them have sum at most 54. By the Pigeonhole Principle, two of these subsets would have the same sum, a contradiction to the givens. | + | But also, <math>S</math> can't have at least 6 elements. To see why, note that at least <math>1 + 6 + \dbinom{6}{2} + \dbinom{6}{3} + \dbinom{6}{4}=57</math> of its subsets have at most four elements (the number of subsets with no elements plus the number of subsets with one element plus the number of subsets with two elements plus the number of subsets with three elements plus the number of subsets with four elements), so each of them have sum at most 54. By the Pigeonhole Principle, two of these subsets would have the same sum, a contradiction to the givens. |
Revision as of 14:36, 3 January 2014
Problem
Let the sum of a set of numbers be the sum of its elements. Let 
be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of have the same sum. What is the largest sum a set with these properties can have?
Solution
The maximum is 
, attained when 
. We must now prove that no such set has sum at least 62. Suppose such a set existed. Then must have more than 4 elements, otherwise its sum would be at most 
if it had 4 elements.
But also, can't have at least 6 elements. To see why, note that at least 
of its subsets have at most four elements (the number of subsets with no elements plus the number of subsets with one element plus the number of subsets with two elements plus the number of subsets with three elements plus the number of subsets with four elements), so each of them have sum at most 54. By the Pigeonhole Principle, two of these subsets would have the same sum, a contradiction to the givens.
Thus, must have exactly 5 elements. contains both 15 and 14 (otherwise its sum is at most 
). It follows that cannot contain both 
and 
for any 
, or the subsets 
and 
would have the same sum. So now must contain 13 (otherwise its sum is at most 
), and cannot contain 12, or the subsets 
and 
would have the same sum.
Now the only way could have sum at least 
would be if 
. But the subsets 
and 
have the same sum, so this set does not work, a contradiction to the givens. Therefore no with sum at least 62 is possible and 61 is indeed the maximum.
See also
| 1986 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
