MATH 240 Midterm: 2002 Math 240 - Spring Exam 2
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No notes, no books, no calculator, no cell phones, no pagers, no electronic devices at all. Solutions will be posted shortly after the exam: www. math. wisc. edu/ miller/m240. 2: (8 pts) let x be a positive real number and n any positive integer. Show that there is a positive integer j such that the distance from jx and some integer is 1 n. Hint: for each j there are integers k and m with 0 m n 1 so that k + m n. Jx k + m + 1 n. 3: (8 pts) one hundred tickets numbered 1, 2, . , 100 are to be sold to 100 di erent people. Eleven prizes are awarded, a grand prize (a scuba diving trip to the caribbean island of bonaire), and 10 consolation prizes (a round-trip bus ticket to milwaukee not including taxes or gratuities). Assume each person can win at most one prize.
