/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 The swimming coach at Wohascum H... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 20

The swimming coach at Wohascum High has a bit of a problem. The swimming pool, like much of the building, is not in good repair, and in fact only three of the lanes are really usable. The school swimming championship is coming up, and while the coach expects only a small turnout for this event it is not at all sure that the number of participants will be divisible by 3 , let alone a power of 3 (which would make it easy to arrange a "single-elimination" format). Instead, in order to choose three participants to compete in the final, the coach intends to have all participants swim in an equal number of preliminary races (of course, that number should be positive). Also, she wants each preliminary race to have exactly three swimmers in it. Finally, she does not want any two particular swimmers to compete against each other in more than one of these preliminary races. Can all this be arranged a. if there are five participants; b. if there are ten participants?

Short Answer

Expert verified
No for both 5 and 10 participants; the requirements can't be met with the given constraints for feasible arrangements.

Step by step solution

01

Identify the Problem Requirements

Given the constraints: 1) Every preliminary race must have exactly three swimmers. 2) Each swimmer must swim in an equal number of races. 3) No two swimmers compete against each other more than once.
02

Test for 5 Participants

For 5 participants, list the total number of races needed if each participant races an equal number of times in races of 3 people. Check if any arrangement satisfies the requirements.
03

Solution for 5 Participants

There isn't a feasible arrangement for 5 participants that satisfies the requirements. Because in every parameter taken, the setup either leads to a conflict where swimmers would need to race each other more than once.
04

Test for 10 Participants

For 10 participants, list the total number of races needed if each participant races an equal number of times in races of 3 people. Check if any arrangement satisfies the requirements.
05

Solution for 10 Participants

There isn't a feasible arrangement for 10 participants that satisfies the requirements either. For the configuration to work, the setup again leads to conflicts similar to the 5-participant scenario.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorial Mathematics
Combinatorial mathematics is a branch of mathematics that focuses on counting, arrangement, and combination of objects. It helps solve problems related to discrete structures and is essential in fields such as computer science, statistics, and optimization. In the context of this exercise, combinatorial mathematics plays a crucial role as we attempt to find a feasible arrangement for preliminary races given the constraints. The idea is to explore different combinations and ensure that each swimmer competes the same number of times without overlapping with the same competitors. This includes understanding permutations and how different groupings can fit within the given rules.
Preliminary Races
Preliminary races are the initial rounds of a competition designed to narrow down the number of participants to a smaller group, who then advance to the final or subsequent rounds. For this exercise, the preliminary races must each have exactly three swimmers. A critical goal is that every participant swims an equal number of times, and no two swimmers compete against each other more than once. This setup ensures fairness and balanced competition. Organizing these races requires a good understanding of combinatorial arrangements to ensure all conditions are met.
Tournament Scheduling
Tournament scheduling is the process of arranging matches or races in a way that meets specific criteria, such as fairness and equal opportunities for all participants. It involves creating a schedule that balances numerous constraints. In the context of the problem, each swimmer must swim an equal number of times, each race must have exactly three swimmers, and no two swimmers should race together more than once. Scheduling for this tournament involves careful planning and checking various configurations to meet these criteria. When participants are not perfectly divisible by the number of races required, creating a balanced schedule becomes quite complex.
Graph Theory
Graph theory is a field of mathematics focusing on graphs, which are structures made up of vertices (nodes) connected by edges (lines). This theory is often used to solve problems related to connectivity and structure. In this exercise, graph theory helps visualize and solve the problem of organizing the swimming races. Each participant can be represented as a vertex, and each race as an edge connecting three vertices (swimmers). The challenge becomes forming a graph where each vertex has equal degrees (equal participation), and no two vertices (swimmers) share an edge (race) more than once. Employing graph theory principles allows us to see possible or impossible configurations, helping identify feasible solutions or proving the lack thereof.

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Most popular questions from this chapter

Start with a circle and inscribe a regular \(n\)-gon in it, then inscribe a circle in that regular \(n\)-gon, then inscribe a regular \(n\)-gon in the new circle, then a third circle in the second \(n\)-gon, and so forth. Continuing in this way, the region (disk) inside the original circle will be divided into infinitely many smaller regions, some of which are bounded by a circle on the outside and one side of a regular \(n\)-gon on the inside (call these "type I" regions) while others are bounded by two sides of a regular \(n\)-gon on the outside and a circle on the inside ("type II" regions). Let \(f(n)\) be the fraction of the area of the original disk that is occupied by type I regions. What is the limit of \(f(n)\) as \(n\) tends to infinity?

Starting with an empty \(1 \times n\) board (a row of \(n\) squares), we successively place \(1 \times 2\) dominoes to cover two adjacent squares. At each stage, the placement of the new domino is chosen at random, with all available pairs of adjacent empty squares being equally likely. The process continues until no further dominoes can be placed. Find the limit, as \(n \rightarrow \infty\), of the expected fraction of the board that is covered when the process ends.

We call a sequence \(\left(x_n\right)_{n \geq 1}\) a superinteger if (i) each \(x_n\) is a nonnegative integer less than \(10^n\) and (ii) the last \(n\) digits of \(x_{n+1}\) form \(x_n\). One example of such a sequence is \(1,21,021,1021,21021,021021, \ldots\), which we abbreviate by ...21021. Note that the digit 0 is allowed (as in the example) and that (unlike in the example) there may not be a pattern to the digits. The ordinary positive integers are just those superintegers with only finitely many nonzero digits. We can do arithmetic with superintegers; for instance, if \(x\) is the superinteger above, then the product \(x y\) of \(x\) with the superinteger \(y=\ldots 66666\) is found as follows: \(1 \times 6=6\) : the last digit of \(x y\) is 6 . \(21 \times 66=1386\) : the last two digits of \(x y\) are 86 . \(021 \times 666=13986\) : the last three digits of \(x y\) are 986 . \(1021 \times 6666=6805986\) : the last four digits of \(x y\) are 5986, etc. Is it possible for two nonzero superintegers to have product \(0=\ldots 00000\) ?

Starting with a positive number \(x_0=a\), let \(\left(x_n\right)_{n \geq 0}\) be the sequence of numbers such that $$ x_{n+1}= \begin{cases}x_n^2+1 & \text { if } n \text { is even } \\\ \sqrt{x_n}-1 & \text { if } n \text { is odd }\end{cases} $$ For what positive numbers \(a\) will there be terms of the sequence arbitrarily close to 0 ?

Let \(f_1, f_2, \ldots, f_n\) be linearly independent, differentiable functions. Prove that some \(n-1\) of their derivatives \(f_1^{\prime}, f_2^{\prime}, \ldots, f_n^{\prime}\) are linearly independent.
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