109年 - 109 國立中山大學_碩士班招生考試_資工系(甲組)：離散數學#105758

科目： 中山◆資工◆離散數學 | 年份：109年 | 選擇題數：0 | 申論題數：13

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所屬科目： 中山◆資工◆離散數學

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申論題 (13)

(a) no container is left empty?

(b) the third container has an even number of balls in it?

2. If a, b ∈ Z⁺, and both are odd, prove that 2|(a² + b²) but 4 (a² + b²).

(a) How many closed binary operations functions f: A X A - A are there?

(b) How many of these closed binary operations are commutative?

4. An auditorium has a seating capacity of 900. How many seats must be occupied to guarantee that at least two people seated in the auditorium have the same first and last initials?

5. In how many ways can 3600 identical envelopes be divided, in package of 25, among five student groups so that each group get at least 150, but not more than 1000, of the envelopes?

(a) each summand must appear an even number of times；

(b) each summand must be even.

7. If ,n ≥ 0, is the unique solution of the recurrence relation = 0, and a₂ = 156/77, = 1628/6336, what is d?

8. (a) How many vertices and how many edges are there in the complete bipartite graphs , where m,n ∈ Z⁺.

(b) If the graph has 72 edges, what is m?

9. For a, b,n ∈ Z⁺ and n ＞ 1, prove that a ≡ b (mod n) gcd(a,n) = gcd(b,n).