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題組內容

7. (Algorithm points) In each of the following, . Solve for relative to the given set , and determine the appropriate Ibig-Oh" form for

(a).

其他申論題

(b) Now consider an chessboard for some fixed . ForI , how many squares are contained in this chessboard?

5. Let be a set of five positive integers the maximum of which is at most 9. Prove that the sums of the elements in all the nonempty subsets of S cannot all be distinct.

6. (a) Fermat's Theorem. If is a prime, prove that for each .

(b) Buler's Theorem. For each n Ezt,n ＞ 1, and each a EZ, prove that if gcd(a,n) = 1, then (a中(n) = 1(mod n).

(a) Apply Dijkstra's algorithm to the graph shown in the following Fig. 1 and determine the shortest distance from vertex b to each of the other vertices in the graph.

(b) Find a shortest path from vertex g to each of the vertices a, b, and c.

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