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Chapter 5: Problem 12

Let triangle \(A B C\) be equilateral, with \(A B=1\). Show that if we select 10 points in the interior of this triangle, there must be at least two whose distance apart is less than \(1 / 3\).

Short Answer

Expert verified
By dividing the equilateral triangle into smaller triangles of side \( 1 / 3 \) and applying the Pigeonhole Principle, we find that there must be at least one pair of points with a distance of less than \( 1 / 3 \) apart.

Step by step solution

01

Divide the triangle

Divide the initial equilateral triangle \( A B C \) into 9 smaller equilateral triangles with sides of length \( 1 / 3 \) by drawing straight lines parallel to the sides of the triangle \(A B C \).
02

Apply the Pigeonhole Principle

Place the ten points into these smaller triangles. Since we have 9 smaller triangles but 10 points, by the Pigeonhole Principle, there must exist at least one smaller triangle that contains more than one point.
03

Triangle Properties

In an equilateral triangle, all sides have the same length. By construction, this length in the small triangles is \( 1 / 3 \). Thus, it is impossible for two points to be in the same triangle and be more than \( 1 / 3 \) apart since \( 1 / 3 \) is the maximum distance in one of the smaller triangles.
04

Final Statement

Therefore, in any selection of 10 points from the given equilateral triangle, there is at least one pair of points with a distance of less than \( 1 / 3 \) apart.

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Most popular questions from this chapter

a) Show that if any 14 integers are selected from the set \(S=\\{1,2,3, \ldots, 25\\}\), there are at least two whose sum is 26 . b) Write a statement that generalizes the results of part (a) and Example 5.45.

a) Let \(S \subset \mathbf{Z}^{+}\). What is the smallest value for \(|S|\) that guarantees the existence of two elements \(x, y \in S\) where \(x\) and \(y\) have the same remainder upon division by 1000 ? b) What is the smallest value of \(n\) such that whenever \(S \subseteq \mathbf{Z}^{+}\)and \(|S|=n\), then there exist three elements \(x, y, z \in S\) where all three have the same remainder upon division by 1000 . c) Write a statement that generalizes the results of parts \((\mathrm{a})\) and \((\mathrm{b})\) and Example \(5.43\).

If \(f: A \rightarrow A\) is any function, prove that for all \(m\), \(n \in \mathbf{Z}^{+}, f^{m} \circ f^{n}=f^{n} \circ f^{m}\). (First let \(m=1\) and induct on \(n\). Then induct on \(m\). This technique is known as double induction.)

a) Write a computer program (or develop an algorithm) to determine the minimum and maximum values in an array \(A[1], A[2], A[3], \ldots, A[n]\) of integers. (Here \(n \in \mathbf{Z}^{+}\) with \(n \geq 2\), and the entries in the array need not be distinct.) b) Determine the worst-case complexity function for the implementation developed in part (a).

Let \(f: X \rightarrow Y\), and for each \(i \in I\), let \(A_{i} \subseteq X\). Prove that a) \(f\left(\bigcup_{i \in I} A_{i}\right)=\bigcup_{i \in I} f\left(A_{i}\right)\). b) \(f\left(\bigcap_{i \in r} A_{i}\right) \subseteq \bigcap_{i \in I} f\left(A_{i}\right)\). c) \(f\left(\bigcap_{i \in I} A_{i}\right)=\bigcap_{i \in I} f\left(A_{i}\right)\), for \(f\) injective.
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