CAT 2025 Slot 2 QA Question 20: If $a, b, c$ and $d$ are integers such that their sum is 46, then the minimum possible value of $(a-b)^2 + (a-c)^2 + (a-d)^2$ is

Question

If $a, b, c$ and $d$ are integers such that their sum is 46, then the minimum possible value of $(a-b)^2 + (a-c)^2 + (a-d)^2$ is

Accepted Answer

Let $x = b-a, y = c-a, z = d-a$.

$x+y+z = (b+c+d) - 3a$
$= (46-a) - 3a = 46 - 4a$

The required expression becomes
$x^2 + y^2 + z^2$

Since $46 - 4a$ is always congruent to $2$ modulo $4$, the closest possible sums to $0$ are $2$ and $-2$.

For three integers with sum $2$ or $-2$, the minimum sum o

If a,b,ca, b, ca,b,c and ddd are integers such that their sum is 46, then the minimum possible value of (a−b)2+(a−c)2+(a−d)2(a-b)^2 + (a-c)^2 + (a-d)^2(a−b)2+(a−c)2+(a−d)2 is