HMMT 二月 2002 · 冲刺赛 · 第 12 题

英文解析

12. This question forms a three question multiple choice test. After each question, thereare 4 choices, each preceded by a letter. Please write down your answer as the ordered triple
    (letter of the answer of Question #1, letter of the answer of Question #2, letter of the answerof Question #3). If you find that all such ordered triples are logically impossible, then write
    “no answer” as your answer. If you find more than one possible set of answers, then provideall ordered triples as your answer.
    When we refer to “the correct answer to Question X ” it is the actual answer, not theletter, to which we refer. When we refer to “the letter of the correct answer to question X ”
    it is the letter contained in parentheses that precedes the answer to which we refer.
    You are given the following condition: No two correct answers to questions on the testmay have the same letter.
    Question 1. If a fourth question were added to this test, and if the letter of its correct answerwere (C), then:
    (A) This test would have no logically possible set of answers.
    (B) This test would have one logically possible set of answers.
    (C) This test would have more than one logically possible set of answers.
    (D) This test would have more than one logically possible set of answers.
    Question 2. If the answer to Question 2 were “Letter (D)” and if Question 1 were not onthis multiple-choice test (still keeping Questions 2 and 3 on the test), then the letter of theanswer to Question 3 would be:
    (A) Letter (B)
    (B) Letter (C)
    (C) Letter (D)
    (D) Letter (A)
    Question 3. Let P = 1. Let P = 3. For all i > 2, define P = P P − P . Which is a
    1 2 i i − 1 i − 2 i − 2
    factor of P ?
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    (A) 3
    (B) 4
    (C) 7
    (D) 9
    Solution: (A, C, D) . Question 2: In order for the answer to be consistent with thecondition, “If the answer to Question 2 were Letter (D),” the answer to this question actuallymust be “Letter (D).” The letter of this answer is (C).
    Question 1: If a fourth question had an answer with letter (C), then at least two answers 3
    would have letter (C) (the answers to Questions 2 and 4). This is impossible. So, (A) must be the letter of the answer to Question 1.
    Question 3: If we inspect the sequence P modulo 3, 4, 7, and 9 (the sequences quicklyibecome periodic), we find that 3, 7, and 9 are each factors of P . We know that letters
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    (A) and (C) cannot be repeated, so the letter of this answer must be (D).