rachel1234

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.

quasi

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.

rachel1234

My teacher gave it to me, so I don't know the source. And I would love to see the special case.

quasi

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.

quasi

Can I ask what is the level of the math class where you got this problem?

rachel1234

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.

quasi

Cool.

Those are nice problems.

SDK

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?

quasi

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.

justforfun573

stop lying the questions are from usamts and you know it. stop cheating
link - http://usamts.org/Tests/Problems_28_3.pdf