MathLabs

Problem 4

Determine, with proof, the greatest number which is the product of positive integers whose sum is 19761976.
Step 1 of 6: Define the product and sum
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In plain words

The optimization can be studied by replacements that preserve the sum while increasing the product.

f(S)=∏x∈Sx,g(S)=∑x∈Sx=1976f(S)=\prod_{x\in S}x,\quad g(S)=\sum_{x\in S}x=1976
Detailed analysis

Let SS be a multiset of positive integers. Write f(S)=∏x∈Sxf(S)=\prod_{x\in S}x for its product and g(S)=∑x∈Sx=1976g(S)=\sum_{x\in S}x=1976 for its sum. We seek the maximum of f(S)f(S).