Weird Math Question

1. Nov 5, 2008

ritwik06

1. The problem statement, all variables and given/known data

There are five positive integers x1,x2,x3,x4,x5 such that:
$$\sqrt{x1-1}+2\sqrt{x2-4}+3\sqrt{x3-9}+4\sqrt{x4-16}+5\sqrt{x5-25}=x1+x2+x3+x4+x5$$
Fine the possible value of:
$$\frac{x1+x2+x3+x4+x5}{2}$$
a) Data insufficient
b)55
c)110
d) I couldnt exactly remember this option... i think it was greater than 110

I guess, the correct answer is (a). 'cause there is only one equation at my disposal. What o u guys say?

2. Nov 5, 2008

Staff: Mentor

I would try to see if I could make educated guesses at some possible values for x1, x2, ..., x5.
The five integers (x1, ...) are integers, so their sum will be an integer.
The numbers inside the radicals have to be perfect squares, so x1 - 1 has to be a perfect square, which means that x1 could be 2 or 5 or 10 or 17 or ...

Similarly x2 - 4 has to be a perfect square, and x3 - 9 has to be a perfect square, and so on.

You're trying to find the possible value of (x1 + x2 + x3 + x4 + x5)/2. One choice is 55, which means that all five numbers add to 110. The other choice is 110, which means that all five add to 220.

See if this approach will get you somewhere.