Problem Solving

coolhabhi wrote:Consider the expression
(a^2 + a + 1) (b^2 + b + 1) (c^2 + c + 1) (d^2 + d + 1) (e^2 + e + 1)
abcde

where a, b, c, d and e are positive numbers. The minimum value of the expression is

To minimize the value we need to minimize the numerator and maximize the denominator.
But if we try to maximize any factor of the denominator (say a), we will end up in increasing the numerator (a² + a + 1) more.

Hence, let us consider in minimizing the numerator only.
Each of the factors of the numerator will be minimized when a, b, c, d, and e will have the minimum possible value which is 1.

Hence, minimum possible value of the expression = (1 + 1 + 1)*(1 + 1 + 1)*(1 + 1 + 1)*(1 + 1 + 1)*(1 + 1 + 1)/1 = 3^5 = 243

The correct answer is E.

Proper Algebraic Method :
To minimize the expression, we need to minimize each of the factors of the expression like (a² + a + 1)/a, (b² + b + 1)/b, ... etc

It is obvious that if we can minimize any of the five factors, all the others can be minimized in the same way. Let us take the first one.

Now, (a² + a + 1)/a = (a + 1 + 1/a)
Hence, we need to minimize (a + 1/a).

Now, (a + 1/a)² = (a - 1/a)² + 4*a*(1/a) = (a - 1/a)² + 4
As, minimum possible value of (a - 1/a)² is zero, minimum possible value of (a + 1/a)² is 4. Hence, minimum possible value of (a + 1/a) is 2. [As a is a positive integer, we are discarding -2]

Hence, minimum possible value of (a² + a + 1)/a is (2 + 1) = 3.

Hence, minimum possible value of the given expression is 3^5 = 243

The correct answer is E.