The intersection points of $m$ balls in $\mathbf{R}^n$ In $\mathbf{R}^n$ , balls of radii $R_1...R_m$ centered at $A_1...A_m$ have a common point . Prove that if $B_1...B_m$ are points such that $|B_iB_j|\le|A_iA_j|$ , then the balls of radii $R_1...R_m$ centered at $B_1...B_m$ also have a common point .
Consider any of the two circles, suppose the one centered at $B_i$ with radius $R_i$ and that at $B_j$ with radius $R_j$. Name them $b_1$ and $b_2$. Translate the point $A_i$ to $B_i$ and rotate the line $A_iA_j$ so that it coincides with $B_iB_j$. Then it becomes obvious that $b_1$ and $b_2$ has a common point. As we have shown this for any two circles, the result follows.
Consider any of the two circles, suppose the one centered at $B_i$ with radius $R_i$ and that at $B_j$ with radius $R_j$. Name them $b_1$ and $b_2$. Translate the point $A_i$ to $B_i$ and rotate the line $A_iA_j$ so that it coincides with $B_iB_j$. Then it becomes obvious that $b_1$ and $b_2$ has a common point. As we have shown this for any two circles, the result follows.