Question 6
Let $n\ge2$. Suppose $x_1\ge\cdots\ge x_n$ and $y_1\ge\cdots\ge y_n$ are real, with $\sum x_i=\sum y_i=0$ and $\sum x_i^2=\sum y_i^2=1$. Prove
$$\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac2{\sqrt{n-1}}.$$
Proposed by David Speyer, Kiran Kedlaya. Source: Evan Chen’s third-party solution notes.
Source video link: https://youtu.be/r7j0oRtpErA?t=6165
$$\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac2{\sqrt{n-1}}.$$
Proposed by David Speyer, Kiran Kedlaya. Source: Evan Chen’s third-party solution notes.
Source video link: https://youtu.be/r7j0oRtpErA?t=6165