HMIC 2019 Problem 3

Do there exist four points $P_{i}=\left(x_{i}, y_{i}\right) \in \mathbb{R}^{2}(1 \leq i \leq 4)$ on the plane such that:

• for all $i=1,2,3,4$, the inequality $x_{i}^{4}+y_{i}^{4} \leq x_{i}^{3}+y_{i}^{3}$ holds, and
• for all $i \neq j$, the distance between $P_{i}$ and $P_{j}$ is greater than 1?