Difference between revisions of "2006 SMT/General Problems/Problem 8"
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<cmath>(a_1,a_2,a_3,a_4) \in \{(1,1,1,1), (2,1,1,1), (3,1,1,1), (4,1,1,1), (2,2,1,1), (5,1,1,1)\}</cmath> | <cmath>(a_1,a_2,a_3,a_4) \in \{(1,1,1,1), (2,1,1,1), (3,1,1,1), (4,1,1,1), (2,2,1,1), (5,1,1,1)\}</cmath> | ||
Revision as of 17:36, 14 January 2020
Problem
Given two 's, "plus" can be changed to "times" without changing the result: . The solution with three numbers is easy too: . There are three answers for the five-number case. Which five numbers with this property has the largest sum?
Solution
Assuming all the numbers are positive, lets denote the five numbers as . Assume WLOG that . The condition we are looking for is .
Manipulating the LHS, we see that:
We now see that we need to find all unordered 4-tuples of positive integers such that the product of all the elements is less that or equal to five. Those tuples are:
We can test each tuple separately, and solve for by plugging them into the original constraint. This yields the solutions
We have three solutions, as said in the problem, therefore the tuple with the greatest sum is clearly . Our solution is