What is equal to 1 / (1+a^(n-m)) + 1 / (1+a^(m-n)) .

Question

Two whole numbers, m and n, have been chosen. Both are greater than 1 and the sum of them is less than 100. The product, m × n, is given to mathematician X. The sum, m + n, is given to mathematician Y. Then both mathematicians have the following conversation:

X: "I have no idea what your sum is, Y."
Y: "That's no news to me, X. I already knew you didn't know that."
X: "Aha! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"

Answer 1

If a = 0

• If m is different from n, then either 0^(n-m) or 0^(m-n) is undefined (you cannot set zero to a negative power).
• If m equals n, then 0^0 is usually defined by convention to 1, and the expression becomes 1 / (1+1) + 1 / (1+1) = 1/2 + 1/2 = 1.

If a is not zero

1 / (1+a^(n-m)) + 1 / (1+a^(m-n))
= (1+a^(m-n) + 1+a^(n-m)) / ( (1+a^(n-m))*(1+a^(m-n)) )
= (2 + a^(m-n) + a^(n-m)) / (1 + a^(n-m) + a^(m-n) +a^(n-m)*a^(m-n))

but: a^(n-m) = a^-(n-m) = 1 / a^(m-n)
thus: a^(n-m)*a^(m-n) = a^(n-m) / a^(m-n) = 1

We can conclude:
1 / (1+a^(n-m)) + 1 / (1+a^(m-n))
= (2 + a^(m-n) + a^(n-m)) / (2 + a^(n-m) + a^(m-n))
= 1

Conclusion

If a=0 and m<>n, the expression is undefined, else it equals to 1.