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馨小妞>試卷(2022/01/14)

中山◆資工◆離散數學題庫 下載題庫

106 年 - 106 國立中山大學_碩士班招生考試_資工系(甲組):離散數學#105791 

選擇:0題,非選:10題 我要補題 回報試卷錯誤
【非選題】
1.
There are 7 problems in this test. No calculators are allowed. Write down detailed steps for the solution to each problem. Otherwise, no credits for that problem will be given.

【題組】

1. Let x1, x2, , 61e133aa10d5e.jpg be a sequence of n integers. A consecutive subsequence of x1,x2,... ., 61e133db17090.jpg is a subsequence61e13404249b2.jpg for some i, j, 1 ≤ i ≤ j ≤n. Show that for any k, I k n, there is a consecutive subsequence whose sum is divisible by k.



【非選題】
2.【題組】

2. Assume that a sequence of numbers is deined by x0 = 0, x1 = 1, and 61e1346a3d6c6.jpg = 61e13478d7dca.jpg > 1. Find generating function for the sequence, and then find an explicit expression for un.



【非選題】
3.【題組】

3. Show that 61e134923516f.jpg Give combinatorial explanation to the equation.



【非選題】
4.

There are 7 problems in this test. No calculators are allowed. Write down detailed steps for
the solution to each problem. Otherwise, no credits for that problem will be given.
4. A bipartite graph is a graph whose vertices can be partitioned into two subsets X and Y so that each edge has one end in X and the other end in Y: A cycle of G is a sequence of vertices u0, u1, ...,61e135c4012ea.jpg such that each vertex 61e135ffe409c.jpgis distinct, except 61e134c816ae6.jpg,and 61e134dec5974.jpg; are adjacent for each i = 1,2,. ,I. The length of the cycle is l. A k-cube is a graph whose vertices are binary strings of length k, for some integer k > 0.Two vertices are adjacent if and only if they differ in exactly one bit.


【題組】(a) Show that every h-cube is bipartite by partitioning its vertexes into X and Y, and then show that every edge connects some vertex in X and another vertex in Y.


【非選題】
5.【題組】(b) Show that a bipartite graph has no cycles of odd length.

【非選題】
6.【題組】(c) Show that if a graph has no cycles of odd length then it is bipartite.

【非選題】
7.
There are 7 problems in this test. No calculators are allowed. Write down detailed steps for the solution to each problem. Otherwise, no credits for that problem will be given.

【題組】5. Suppose that 5 points are chosen in a square whose sides have length 2. Show that there must be at least two points p and g such that the distance between them is no more than √2.


【非選題】
8.【題組】

6. Fibonacci numbers are deined as f0 = 0, f1= 1 and 61e13651a6648.jpg for n >.1. Show that 61e1367566554.jpg is even, for every positive integer k. .



【非選題】
9.
There are 7 problems in this test. No calculators are allowed. Write down detailed steps for the solution to each problem. Otherwise, no credits for that problem will be given. 
7. Let m and n be two positive integers, m ≤ n. Define 61e136ad50c63.jpg

【題組】

(a) Show that if'n is prime, then n divides 61e136c41a947.jpgfor every i, 1 ≤i<n.



【非選題】
10.【題組】

(b) Show that if n is composite, then n does not divide 61e136eb9f44b.jpgfor some i, 1 ≤i <n.



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106 年 - 106 國立中山大學_碩士班招生考試_資工系(甲組):離散數學#105791-阿摩線上測驗

106 年 - 106 國立中山大學_碩士班招生考試_資工系(甲組):離散數學#105791