December 31st, 2016, 08:43 AM  #1 
Newbie Joined: Dec 2016 From: New Jersey Posts: 7 Thanks: 1  Points on a plane
Let A1, ..., An and B1,..., Bn be sets of points on a plane. Suppose for all points x, D(x, A1) + D(x, A2) + ... + D(x, An) ≥ D(x, B1) + D(x, B2) + ... + D(x, Bn) , where D(x, y) signifies the distance between x and y. Show that the Ai’s and the Bi’s share the same center of mass. NOTE: A1, An, B1, Bn, A2, B2, Ai, and Bi, are all supposed to have subscripts for the second letters. 
January 1st, 2017, 06:23 AM  #2 
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 
Nice problem. I don't have a solution at this point. Can you reveal the source of the problem? Update: I can prove it for the special case where all points lie on the same line. Last edited by quasi; January 1st, 2017 at 06:40 AM. 
January 1st, 2017, 07:13 AM  #3 
Newbie Joined: Dec 2016 From: New Jersey Posts: 7 Thanks: 1  My teacher gave it to me, so I don't know the source. And I would love to see the special case.

January 1st, 2017, 12:30 PM  #4 
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 
The onedimensional case is easy and fun  you should try it. Here's the statement to prove: If $a_1,\ldots ,a_n$ and $b_1,\ldots ,b_n$ are real numbers such that $$xa_1+\cdots +xa_n \ge xb_1+\cdots +xb_n$$ holds for all real numbers x, then $a_1+\cdots +a_n = b_1+\cdots +b_n$. Hint: First choose x to the left of all the $a$'s and $b$'s. The given inequality now simplifies to what statement about the $a$'s and $b$'s? Next choose x to the right of all the $a$'s and $b$'s, and answer the same question. Last edited by quasi; January 1st, 2017 at 12:40 PM. 
January 1st, 2017, 12:35 PM  #5 
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 
Can I ask what is the level of the math class where you got this problem?

January 1st, 2017, 05:04 PM  #6 
Newbie Joined: Dec 2016 From: New Jersey Posts: 7 Thanks: 1  
January 1st, 2017, 05:32 PM  #7 
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 
Cool. Those are nice problems. 
January 1st, 2017, 09:44 PM  #8 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
Suppose n = 1. Can you prove it in this case? What does it mean for $A_1,B_1$ to have the same center of mass? Now, assume inductively that it is true whenever $1 \leq n \leq N$. Consider a pair of collections of size $N+1$. If this pair satisfies the hypothesis, then for every $x$ in the plane, the given inequalities hold. Can you think of a "correct" choice for $x$ which allows you to apply the induction hypothesis? 
January 1st, 2017, 10:25 PM  #9 
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 
I don't see it. The verification for n = 1 is instant. But I don't see how induction gets you from n = 1 to n = 2. 
January 2nd, 2017, 06:39 PM  #10  
Newbie Joined: Jan 2017 From: Georgia Posts: 10 Thanks: 3  Quote:
link  http://usamts.org/Tests/Problems_28_3.pdf  

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calculus, hard, plane, points 
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