Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?  Select the correct option.
A) The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 – 2m + 1 B) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 – 2m + 1. C) The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1 D) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1 E) None
Explanation :   Given,   =&gt; a + b + c + d = 4m + 1. now squaring this equation:   =&gt; (a+b+c+d)2 = 16m2 + 8m + 1.   =&gt; a2 + b2 + c2 + d2 + 2*( ab + ac + ad + bc + bd + cd ) = 16m2 + 8m + 1.   using the fact that product of two numbers is maximum when they are equal / nearly equal,   expression ab + ac + ad + bc + bd + cd will be maximum when ab , ac , ad, bc , bd , cd will be maximum and this is possible when a, b, c, d are nearly equal to each other. i.e. m , m , m , m+1.   when ab + ac + ad + bc + bd + cd is maximum , obviuosly , a2 + b2 + c2 + d2 will be minimum.   Put values in the above equation we get that the minimum value of a2 + b2 + c2 + d2 comes out to be 4m2 + 2m + 1.   Answer : The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1
<result>C</result>
Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?