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首頁 > 離散數學 > 109年 - 109 教育部公費留學考試：離散數學#125735 > 申論題

題組內容

2. Answer the following two questions.

(a) What is the coefficient of in the expansion of ?

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(b) Use the formula from (a) to prove

(b) Use the result of (a) to compute the coefficient of in the expansion of .

3. Prove or disprove the following statement. Given a bipartite graph, if the number of vertices of odd degree on the left subgraph (subgraph A below) is odd, then the number of vertices of odd degree on the right subgraph (subgraph B below) is also odd.

4. Simplify the Boolean formula tr(M²) given the matrix M below with as few operations (AND, OR) as possible. The trace of a square matrix, denoted as tr(*) is defined by the summation of all diagonal elements in the matrix.

5. Convert the following finite state machine to another machine where every two transitions becomes one. That is, we have two alphabets, one after another in each transition in the new machine while all the states remain to be the same to the original machine. We assume the starting state is A.

6.(15%) Let n be a positive integer greater than or equal to 3. Find the number of subgraphs of an n-vertex complete graph Kn that is isomorphic to a cycle with n vertices.

5.(20%) Find the coefficient of x"in the following generating function: [ (1- 2x)(1- 3x) ]-1

4.(15%) Let gcd(x, y) denote the greatest common divisor of x and y. Find thenumber of integers k ∈ [1, 720] with gcd(k, 720) ＞ 1.

3.(15%) Let n be a positive integer. Show that any n(n - 1) + 1 integers must contain n numbers whose sum is divisible by n.

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