Prove that all terms in an arithmetical equations are equals with border conditions

Would it be possible to prove that there is an equation that includes a number N of unknown numbers that are all equal, between 0 and 1 and whose sum is equal to 1 ? And to find this equation ?

I don't know if the problem is well defined and if it's possible. It works with A - B = 0 with A = B = 0.5 but I'm struggling to prove and extend it. I'm not sure that it would be "legal" to do any multiplication or division.

Regards

• Why does $A-B=0$ force $A=B=\frac 12$? That is one solution, but there are many more that do not have $A+B=1$ – Ross Millikan May 14 at 15:23

You can certainly do $$\left(a-\frac 13\right)^2+\left(b-\frac 13\right)^2+\left(c-\frac 13\right)^2=0$$ with the obvious generalization to $$n$$ variables.
• We square them because squares are always greater than or equal to zero. If a sum of squares is zero, each term must be zero. That is how I get away from the problem of having one equation in more than one variable-really I have $n$ equations because each term must be zero. Can you see the connection between the fact that I have three terms and the fractions are $\frac 13$? That shows how to do it with $n$ variables. – Ross Millikan May 15 at 5:53