Minimum Value of Sum of Squares

CAT 2017 Slot 1 · QA · Hard · Algebra

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If a,b,c,a, b, c, and dd are integers such that a+b+c+d=30a + b + c + d = 30, then the minimum possible value of (a−b)2+(a−c)2+(a−d)2(a - b)^2 + (a - c)^2 + (a - d)^2 is

Answer

2

Explanation

To minimize (a−b)2+(a−c)2+(a−d)2(a - b)^2 + (a - c)^2 + (a - d)^2, a,b,c,da, b, c, d should be as close as possible. Average of 30 among 4 numbers is 7.57.5. So choices for (a,b,c,d)(a, b, c, d) close to 7.5: let a=8,b=8,c=7,d=7a = 8, b = 8, c = 7, d = 7. Then a+b+c+d=30a + b + c + d = 30. Value =(8−8)2+(8−7)2+(8−7)2=0+1+1=2= (8 - 8)^2 + (8 - 7)^2 + (8 - 7)^2 = 0 + 1 + 1 = 2.

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