Find all non-negative real numbers x_i where i \in \{1,2,3,...,n\} such that the following holds for positive integer n:
x_1+x_2^2+x_3^3+...+x_n^n=n
x_1+2x_2+3x_3+...+nx_n=\frac{n(n+1)}{2}
x_{1} + x_{2}^{2} + \dots + x_{n}^{n} = n
\;\;\Longleftrightarrow\;\;
x_{1} + (x_{2}^{2}+1) + \dots + \Big(x_{n}^{n} + \underbrace{1 + 1 + \dots + 1}_{n-1 \text{ times}}\Big)
= n + \frac{(n-1)n}{2} = \frac{n(n+1)}{2}.
Now, applying the AM-GM inequality in each bracket, we get
\frac{n(n+1)}{2} = x_{1} + (x_{2}^{2}+1) + \dots + \Big(x_{n}^{n} + \underbrace{1 + 1 + \dots + 1}_{n-1 \text{ times}}\Big)
\ge
x_{1} + 2x_{2} + \dots + n x_{n} = \frac{n(n+1)}{2}.
Thus, equality must hold in each bracket, which gives
x_{1} = x_{2} = \dots = x_{n} = 1,
which works.
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