If a^2/(b + c) = b^2/(c + a) = c^2/(a + b) = 1 for real numbers a, b and c, then what is the value of the sum 2/(1 + a) + 2/(1 + b) + 2/(1 + c)?

Introduction / Context:
This problem combines symmetry and simple algebraic manipulation. The equal fractions involving a^2, b^2, and c^2 suggest that a, b, and c share a common pattern. Once we find specific values consistent with the given condition, we can evaluate the required sum. Problems like this appear frequently in algebra and aptitude tests to encourage recognition of symmetry and substitution strategies.

Given Data / Assumptions:

a^2/(b + c) = b^2/(c + a) = c^2/(a + b) = 1.
a, b, and c are real numbers, and none of the denominators b + c, c + a, a + b is zero.
The required sum is 2/(1 + a) + 2/(1 + b) + 2/(1 + c).

Concept / Approach:
From a^2/(b + c) = 1 we immediately obtain a^2 = b + c. Similarly, b^2 = c + a and c^2 = a + b. This is a symmetric system of three equations. Often such a system has a simple solution where all variables are equal. Testing a = b = c is an efficient first step. If a = b = c, the equations reduce to a single equation, which can be solved easily. Once a, b, and c are known, evaluating the required sum is straightforward.

Step-by-Step Solution:
From a^2/(b + c) = 1 we have a^2 = b + c.From b^2/(c + a) = 1 we have b^2 = c + a.From c^2/(a + b) = 1 we have c^2 = a + b.Assume a = b = c due to symmetry and test this possibility.If a = b = c, then a^2 = b + c becomes a^2 = a + a = 2a.So a^2 − 2a = 0 gives a(a − 2) = 0, hence a = 0 or a = 2.If a = 0, then b and c are also 0, but this makes denominators like a + b equal to 0, which is not allowed.Therefore a = b = c = 2 is the valid solution.Now evaluate the required sum: 2/(1 + a) + 2/(1 + b) + 2/(1 + c) = 3 * 2/(1 + 2) = 3 * (2/3) = 2.

Verification / Alternative check:
Substitute a = b = c = 2 back into the original equalities: a^2/(b + c) = 4/(2 + 2) = 4/4 = 1, and similarly for b and c. This confirms that a = b = c = 2 satisfies all the given conditions. The function 2/(1 + a) + 2/(1 + b) + 2/(1 + c) is fully symmetric in a, b, and c, so evaluating it at the symmetric solution is sufficient for this problem.

Why Other Options Are Wrong:
Values 0, 1, 3, and 6 would correspond to different specific values of a, b, and c. However, under the constraint a^2 = b + c, b^2 = c + a, and c^2 = a + b, the only non trivial symmetric solution that keeps denominators non zero is a = b = c = 2, producing a sum equal to 2 and no other value in the options.

Common Pitfalls:
Some students attempt to solve the three equations simultaneously in a completely general way, which is algebraically heavy and unnecessary. Others may forget to check the case a = b = c or ignore the requirement that denominators must not be zero. Recognizing symmetry and testing simple equal solutions is an important problem solving strategy for such questions.

Final Answer:
The value of 2/(1 + a) + 2/(1 + b) + 2/(1 + c) is 2.