P A_1 + P A_2 + ... + P A_{100} >= 100, A_i in unit circle

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P A_1 + P A_2 + ... + P A_{100} >= 100, A_i in unit circle

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#1 h Apr 12, 2021, 1:13 PM • 1 Y

Y by Mango247

Inside the circle of radius $1$ there are points $A_1, A_2, A_3, ..., A_{100}$ . Prove that there is one point $P$ at the edge of the circle for which $P A_1 + P A_2 + ... + P A_{100} \ge 100$ .

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#2 h Apr 12, 2021, 1:43 PM • 1 Y

Y by Smileyklaws

Consider two antipodes of the circle, $P$ and $Q.$ By the triangle inequality, $\sum_{i=1}^{100} PA_i + A_iQ \ge 100 PQ = 200.$ Therefore, $\left(\sum_{i=1}^{100} P A_i \right)+ \left(\sum_{i=1}^{100} Q A_i\right) \ge 200,$ so one of these two sums is at least $100$ as desired.

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