<s>[INST] For positive $x,y$ such that $x+y \leq x^2+y^2$ prove inequality for every real $t \ge 1$ $x^t+y^t \leq x^{t+1}+y^{t+1}$. 
For positive $x,y$ such that $x+y \leq x^2+y^2$ prove inequality for every real $t \ge 1$ $x^t+y^t \leq x^{t+1}+y^{t+1}$.

I see that $\frac{(x+y)}{2} \leq \frac {x^{t+1}+y^{t+1}}{(x^t+y^t)}$
 [/INST] It is enough to show that for $x, y > 0$:
$$(x^t+y^t)(x^2+y^2) \le (x^{t+1}+y^{t+1})(x+y)$$
$$\iff x^t+y^t\ge x^{t-1}y+y^{t-1}x$$
which follows from Rearrangement inequality as $x \ge y \iff x^{t-1} \ge y^{t-1}$ for $t \ge 1$.
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