One of my friends who knows I like math puzzles (but doesn't know that optimization problems are sort-of my weakness), gave me the following problem:
Given that:
2(10a + 13b + 14c + 15d) − ((a^2 + b^2 + c^2 + d^2)/3) = 2020,
what is the maximum possible value of a + b + c + d.
I don't really know where to begin on this problem. I tried contacting the friend for help/a hint a couple times last week, but she still hasn't gotten back to me. I was wondering if someone could help me figure out how to solve this problem?
Any help would be greatly appreciated. Thanks in advance!
Hint: What you are given is equivalent to
$$ ( a - 30)^2 + (b-39)^2 + (c-42)^2 + (d-45)^2 = 150 .$$
From there, what is the maximum value of $ ( a - 30) + (b-39) + (c-42) + (d-45)$?
Hence, conclude that the answer is $156 + 10 \sqrt{6} $.