Q7
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
Answer: 2
Key idea: The expression is smallest when \(b, c, d\) are as close to \(a\) as possible.
Write \(b = a+x\), \(c = a+y\), \(d = a+z\) with integers \(x, y, z\). The condition becomes \(4a + x + y + z = 46\) and the expression is \(x^2 + y^2 + z^2\).
- Value 0 needs \(x=y=z=0\), so \(4a = 46\): not an integer.
- Value 1 needs exactly one of \(x,y,z\) to be \(\pm1\), so \(4a = 45\) or \(47\): impossible.
- Value 2: take \(a = 11\), \(x = y = 1\), \(z = 0\), i.e. \((a,b,c,d) = (11,12,12,11)\). Sum \(= 46\) and the expression \(= 1 + 1 + 0 = 2\).
Hence, the answer is 2.