Solve Weird Math Question: Find Value of x1+x2+x3+x4+x5/2

[ritwik06]

Homework Statement

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 couldn't 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?

 

[Mark44]

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.

 

[Architeuthis Dux]

I would say that the correct answer is indeed (a) - data insufficient. The given equation only provides one constraint on the five unknown variables, making it impossible to determine a unique value for (x1+x2+x3+x4+x5)/2. Without additional information or equations, we cannot solve this problem and determine the value of (x1+x2+x3+x4+x5)/2.