# Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$?

My Attempt: Let there are $n$ real numbers and we call them $x_{1},x_{2},..,x_{n}$. Since they are positive real so WLOG we can assume $x_{1}\geq x_{2}..\geq x_{n}$. The condition $x_{1}+x_{2}+..+x_{n}=n \implies x_{n} \leq \frac{1}{n}$. Also $x_{1}\geq x_{2}..\geq x_{n} \implies x_{1}^{2}\geq x_{2}^{2}..\geq x_{n}^{2}$
After this point I am stuck.

• How can the sum of several positive integers be equal to $1$? – Jimmy R. Mar 26 '16 at 12:35
• @JimmyR Thanks for the hint. – rugi Mar 26 '16 at 12:50