tut10a-large - Problems to Week 11 Tutorial — MACM 101 1....

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Unformatted text preview: Problems to Week 11 Tutorial — MACM 101 1. Determine the coefficient of x9 y 3 in the expansion of (a) (x + y )12 , (b) (x +2y )12 , (c) (2x − 3y )12 , (d) (x − y )14 . 2. Show that for all positive integers m and n, n· m+n m = (m + 1) · m+n m+1 . 3. With n a positive integer, evaluate the sum n 0 −3 n 1 + 32 n 2 + . . . + (−1)n 3n n n . 4. For every positive number n, show that n 0 + n 2 + n 4 +. . . = n 1 + n 3 + n 5 +. . . 5. Show that if any 14 numbers are selected from the set S = {1, 2, 3, . . . , 25}, there are at least two whose sum is 26. 6. Let triangle ABC be equilateral, with AB = 1. Show that if we select 10 points in the interior of this triangle, there must be at least two whose distance apart is less than 1/3. 7. During the first six weeks of his senior year in college, Brace sends out at least one resume each day but no more than 60 resumes in total. Show that there is a period of consecutive days during which he sends out exactly 23 resumes. ...
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