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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. |
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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. |
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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.
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January 1st, 2017, 12:30 PM | #4 |
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 |
The one-dimensional 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 $$|x-a_1|+\cdots +|x-a_n| \ge |x-b_1|+\cdots +|x-b_n|$$ 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. |
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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?
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January 1st, 2017, 05:04 PM | #6 |
Newbie Joined: Dec 2016 From: New Jersey Posts: 7 Thanks: 1 | |
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January 1st, 2017, 05:32 PM | #7 |
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 |
Cool. Those are nice problems. |
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January 1st, 2017, 09:44 PM | #8 |
Senior Member Joined: Sep 2016 From: USA Posts: 357 Thanks: 196 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? |
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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. |
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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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