Uzumaki's impossible equality

Note that we must have $x_i \ge i^2$ for all $i > 0$ . Then the AM-GM inequality gives $\frac{x_i}{2} = \frac{(x_i-i^2) + i^2}{2} \ge \sqrt{(x_i-i^2)i^2} = i \sqrt{x_i-i^2},$ or equivalently, $x_i \ge 2i \sqrt{x_i - i^2},$ with equality occurring if and only if $i^2 = x_i - i^2$ , or $x_i = 2i^2$ . Therefore, the LHS sum is always at least as large as the RHS sum, except when $x_i = 2i^2$ for each $i = 1, 2, \ldots, 13$ ; therefore, $x_{13} = 2(13)^2 = \boxed{338}$ .

Make the substitution $u_i = \sqrt{x_i -i^2}$ , equivalently $x_i=i^2 + u_i ^2$ . Then the $u_i$ 's satisfy $\sum u_i ^2 - 2i u_i + u_i ^2 = \sum (u_i -i)^2 = 0$ . It follows that $\forall i , u_i =i$ , hence $x_{13} = 13^2 + 13^2 = \boxed{338}$

In the solution, will use the inequality $2ab \leq a^2+b^2$ , with the equality sign occurring if and only if $a=b$ .

Now, for a fixed value of $i$ , we have $2i\sqrt{x_i-i^2} \leq i^2 + (x_i-i^2) = x_i$ Thus adding all the terms corresponding to $i=1$ to $i=13$ would give

$2\sum_{i=1}^{13} i\sqrt{x_i-i^2} \leq \sum_{i=1}^{13}x_i$

The equality sign occurs if and only if $i=\sqrt{x_i-i^2}$ , for all $i$ running from $1$ to $13$ . This means $i^2 = x_i-i^2$ , or $x_i = 2i^2$ for all $i$ from $1$ to $13$ .

Therefore $x_{13} = 2(13^2) = \fbox{338}$

Consider the general form $\sum_{i=1}^{n}x_i=2\sum_{i=1}^{n}i\sqrt{x_i-i^2}=0.$ Moving all the terms to the LHS yields $\sum_{i=1}^{n}\left(x_i-2i\sqrt{{x_i-i^2}}\right)=\sum_{i=1}^{n}\left(\sqrt{x_i-i^2}-i\right)^2=0,$ from which we can surmise that $x_i=2i^2.$ Thus, we have that $x_{13} = 2 \cdot 13^2 = \boxed{338},$ as desired.

$\sqrt{x_{1}-1^{2}}+2\sqrt{x_{2}-2^{2}}+3\sqrt{x_{3}-3^{2}}+...+13\sqrt{x_{13}-13^{2}}=\frac{1}{2}(x_{1}+x_{2}+x_{3}...+x_{13})$ $\Rightarrow 2\sqrt{x_{1}-1^{2}}+4\sqrt{x_{2}-2^{2}}+... +2n\sqrt{x_{13}-13^{2}}=x_{1}+x_{2}+...+x_{13}\\ \Rightarrow x_{1}+x_{2}+...+x_{13}-2\sqrt{x_{1}-1^{2}}-4\sqrt{x_{2}-2^{2}}-...-26\sqrt{x_{13}-13^{2}}=0$ $\Rightarrow \left ( \sqrt{ x_{1}-1}-1\right )^{2}+\left ( \sqrt{ x_{2}-2^{2}}-2\right )^{2}+...+\left ( \sqrt{ x_{13}-13^{2}}-13\right )^{2}=0$ \ we have: $\sqrt{x_{1}-1}-1=0 \Rightarrow \sqrt{x_{1}-1}=1 \Rightarrow x_{1}-1=1 \Rightarrow x_{1}=2\\ \sqrt{x_{2}-2^{2}}-2=0 \Rightarrow \sqrt{x_{2}-2^{2}}=2 \Rightarrow x_{2}-2^{2}=4 \Rightarrow x_{1}=2.2^{2}\\ \vdots\\ \sqrt{x_{13}-13^{2}}-13=0 \Rightarrow \sqrt{x_{13}-13^{2}}=13 \Rightarrow x_{13}-13^{2}=13^{2} \Rightarrow x_{13}=2.13^{2}=338$

The quick and dirty way of getting at the answer to this one is to assume that if any sequence ${ x }_{ 1 }, { x }_{ 2 },...{ x }_{ 13 }$ to make this equality true, then
${ x }_{ 13 }$ must be the answer. So, let ${ x }_{ i }=2{ i }^{ 2 }$ . Then this equality is obviously true. Thus, ${ x }_{ i }=2{ (13) }^{ 2 }=338$ .

Since, the $13$ real numbers are independent of each other, and seperable from each other in terms of the corresponding $13$ real nos.. Therefore, we can seperate out the entities corresponding to each of the $13$ real nos. Since, we need to find $x_{13}$ , therefore, for the given expression, the expression corresponding only to $x_{13}$ and independent to other terms, can be written as $x_{13}=2\times13\times\sqrt{x_{13}-13^2}$ which can be further simplified into, $\Rightarrow(x_{13}-338)^2=0$ Thus, giving the required values of $x_{13}$ as $\{338,338\}$ . $\text{Hence, the required value of }x_{13}=\fbox{338}$