Equal 1st and 2nd degree sums implies equal entries

Question

Let's say we have 2 sets of entries $x_i$ and $y_i$ for $i=1(1)n$. We are given two facts are told that they're true. Which are:

1. $\Sigma x_i = \Sigma y_i$, i.e. their linear/first order sums are equal.
2. $\Sigma {x_i}^{2} = \Sigma {y_i}^{2}$, i.e. their second order sums are equal.

Can we therefore conclude that $x_i=y_i$ for each $i$?

Actually, this is a bit imprecise. A better way to present my question would be to ask: Can we therefore say, $$\Bigl( X= \{ x_i\mid i=1(1)n \} \Bigr) = \Bigl( Y= \{ y_i\mid i=1(1)n \} \Bigr) ?$$

So my question is this: If 2 sets of entries have the same 1st and 2nd order sums, should the 2 sets be equal?

Extended Discussion:

1. If we take some other pair of consecutive orders, such as 3rd and 4th, and say that the sums of the 2 sets of entries in each those respective orders are equal, then should our entry sets be, again, equal?
2. What if we take non consecutive orders? Such as 10th and 15th?
3. What if we take fractional orders?

P.S.: I might've tagged irrelevant topics for this question. I'm not sure what this question falls under. Any help will be acknowledged.

Answer 1

Call your first multiset of numbers $\{x,y,z\}$. Then without loss of generality the second multiset will be $\{x+\delta_1,y+\delta_2,z-\delta_1-\delta_2\}$.

Simplify the equation which equates their sums of squares.

You will get a linear equation in $x,y,z$.

Pick any two values for $\delta_1,\delta_2$.

You then just need to solve a single linear equation in $3$ variables.

Just make sure you are not unlucky enough to pick values so that you end up with the same set as you started with, in a different order. You have so much freedom that you can avoid this easily.