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107年 - 107 教育部公費留學考試：離散數學#125745
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三、 Solve the recurrence relation an =
= 1 and a₁ = 2.
其他申論題
(b) Prove that in this case, for every seating arrangement, there is always some person both of whose neighbors are boys.
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(a) Determine whether there exists a connected planar graph that contains 13 edges and 9 regions. If so, draw the graph and prove its correctness. Otherwise, prove there is no such a graph.
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(b) Determine whether there exists a connected planar graph that contains 7 vertices, without an Euler cycle, and with the chromatic number equal to 3. If so, draw the graph and prove its correctness. Otherwise, prove there is no such a graph.
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(c) Let G be a graph with 5 vertices. Prove that if every vertex of G has degree 2, then G must be a cycle.
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四、Suppose that a spam filter is trained from 7500 messages, where 5000 messages among them are spam. The word "surprise" appears in 750 spam messages and 10 non-spam messages, and the word "energy" appears in 400 spam messages and 100 non-spam messages. What is the probability that an incoming message is considered spam by the filter if it contains both the words "surprise" and "energy"?
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五、 Compute , where n ≥ m ≥ 1.
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六、Solve the system of congruences x = 5 (mod 6), x = 3 (mod 10), x = 8 (mod 15), and x = 11 (mod 21).
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1.(1) We know that (a) P(1,1) = 2 (b) P(m+1,n)=P(m,n)+2(m+n) (c) P(m,n+1)=P(m,n)+2(m+n-1) Prove or disprove the following formula for any positive integers m and n: P(m,n) = (m+n)(m+n-1) - 2n +2
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(2) Let f(x) be the ceiling function and let g(x) be the floor function. Let x and y be positive real numbers. Find all conditions of x and y so that f(x+y) = f(x) + g(y) is true.
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2.(1) What is the value of ( mod 17) where mod is the modular operator?
#534346
= 1 and a₁ = 2.
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= 1 and a₁ = 2.