Let a,b,c,d be the four integers such that a+...
Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?

A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1
B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1

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 Mahinder Singh May 22, 2021
Related Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1Explain this question please?
For any expression of the form a+b+c+d, the minimum value of a^2 + b^2+c^2+ d^2 will be when a = b = c = d. Or in other words, the four numbers have to be as close to the average as much as is possible. As the sum is in the form of 4m+ 1, we can say that the four numbers would give the minimum value of the above expression if they are m, m, m and m+1.

So, we get the sum of their squares to be:

4m^2 + 2m + 1 and so, option b is correct.

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