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December 31st, 2016, 09:43 AM   #1
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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, 07:23 AM   #2
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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.
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Last edited by quasi; January 1st, 2017 at 07:40 AM.
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January 1st, 2017, 08:13 AM   #3
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Quote:
Originally Posted by quasi View Post
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.
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, 01:30 PM   #4
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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.
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Last edited by quasi; January 1st, 2017 at 01:40 PM.
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January 1st, 2017, 01:35 PM   #5
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Can I ask what is the level of the math class where you got this problem?
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January 1st, 2017, 06:04 PM   #6
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Can I ask what is the level of the math class where you got this problem?
I'm currently in Calculus AB, but the question wasn't from the class. It was from my math team teacher. She just gave it to us for fun.
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January 1st, 2017, 06:32 PM   #7
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Cool.

Those are nice problems.
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January 1st, 2017, 10:44 PM   #8
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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, 11:25 PM   #9
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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, 07:39 PM   #10
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Quote:
Originally Posted by rachel1234 View Post
My teacher gave it to me, so I don't know the source. And I would love to see the special case.
stop lying the questions are from usamts and you know it. stop cheating
link - http://usamts.org/Tests/Problems_28_3.pdf
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