Consider the unit sphere in \( \mathbb{R}^n \). Find the maximum number of points on the sphere such that the (Euclidean) distance between any two of these points is larger than \( \sqrt 2 \).

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Consider the unit sphere in \( \mathbb{R}^n \). Find the maximum number of points on the sphere such that the (Euclidean) distance between any two of these points is larger than \( \sqrt 2 \).

Let \(P_1,P_2,\ldots,P_n\) be n points in {(x,y): 0<x<1, 0<y<1} (n>1). Let \(r_i=\min_{j\neq i} d(P_i,P_j)\) where d(x,y) means the distance between two points x and y. Prove that \(r_1^2+r_2^2+\cdots+r_n^2\le 4\).