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# 数学在线辅导|Stanford大学数学夏令营测试辅导

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## 下面是几道典型的数学竞赛代写测试题目

An equilateral has sides of length $1 \mathrm{~cm}$.
(a) Show that for any configuration of five points on this triangle (on the sides or in the interior), there is at least one pair of from these five points such that the distance between the two points in the pair is less than or equal to $.5 \mathrm{~cm}$.
(b) Show that $.5$ (in part (a)) cannot be replaced by a smaller number even if there are 6 points.
(c) If there are eight points, can $.5$ be replaced by a smaller number? Prove your answer.
Suppose $n$ is a positive integer. The (imaginary) sea of Babab has islands each of which has an $n$-letter name that uses only the letters ” $\mathrm{a}$ ” and “b,” and such that for each $n$-letter name that uses only the letters “a” and ” $\mathrm{b}$,” there is an island. For example, if $n=3$, then Aaa, Aab, Aba, Baa, Abb, Bab, Bba and Bbb are the islands in the sea of Babab. The transportation system for Babab consists of ferries traveling back and forth between each pair of islands that differ in exactly one letter. For example, there is a ferry connecting Bab and Bbb since they differ only in the second letter.
a) How many islands and how many ferry routes are there in terms of $n$ ? Count the ferry route for both directions as a single ferry route, so for example, the ferry from Bab to Bbb is the same ferry route.
Babab does not have much in the way of natural resources or farm land so nearly all food and supplies are provided by the Babab All Bulk Company (BABCO). The people of Babab (Bababians) desire easy access to a BABCO store, where “easy access” means there is a BABCO store on their own island or on one that they can get to with a single ferry ride. However, BABCO finds it uneconomical to give the people on one island easy access to two different BABCO stores, and BABCO is willing to deny some Bababians easy access to a BABCO store in order to meet this restriction.
b) In the cases $n=3, n=4$, and $n=5$, what is the maximum number of stores that $\mathrm{BABCO}$ can build while satisfying the restriction than no one has easy access to more than one BABCO store? Be sure to prove your answer is optimal.
c) Now suppose BABCO changes its strategy and decides it wants to be sure every Bababian has access to a $B A B C O$ store even if it means some Bababians have easy access to two stores. What is the minimum number of stores needed to satisfy this condition in the cases $n=3, n=4$, and $n=5$ ?
d) Can you find optimal solutions to parts b and $\mathrm{c}$ for $n=6$ ?

## 如何成为一个有竞争力的SUMaC候选人？

PSAT高分也有助于你的申请。与所有标准化考试一样，实践出真知。确保在你的PSAT考试日期之前进行几次模拟测试。

• 高的GPA，包括但不限于数学课程的高成绩
• 高标准的分数，特别是数学部分的分数
• 通过数学竞赛等课外活动表现出对数学的热情
• 参加过以前的数学训练营
• 特别是：在SUMaC基于证明的入学考试中表现优异！