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: 167 Thanks: 79 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  

Tags 
calculus, hard, plane, points 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Find all points on surface whose tangent plane is parallel to given plane  Addez123  Calculus  4  November 12th, 2016 06:51 AM 
2 Points in R3 and 1 point on plane. How to find equation of plane?  extreme112  Linear Algebra  2  October 13th, 2015 07:34 AM 
Points on a plane  wnzguitar  Algebra  2  March 27th, 2011 06:11 AM 
Equation for Plane given 3 points  Anonymouse7  Calculus  1  January 27th, 2010 10:38 AM 
Points in a plane  aman_cc  Applied Math  5  November 21st, 2009 01:06 AM 