By Tilla Weinstein

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**Example text**

M} such that f (A ∩ B) = min{f (A), f (B)}. Prove that m jn. c(n, m) = j=1 (page 173) A4. Let x1 , x2 , . . , x19 be positive integers each of which is less than or equal to 93. Let y1 , y2 , . . , y93 be positive integers each of which is less than or equal to 19. Prove that there exists a (nonempty) sum of some xi ’s equal to a sum of some yj ’s. (page 174) A5. Show that −10 −100 x2 − x 3 x − 3x + 1 1 11 2 dx + 1 101 x2 − x 3 x − 3x + 1 is a rational number. 11 10 2 dx + 101 100 x2 − x 3 x − 3x + 1 2 dx (page 175) A6.

D9 has nine (not necessarily distinct) decimal digits. The number e1 e2 . . e9 is such that each of the nine 9-digit numbers formed by replacing just one of the digits di in d1 d2 . . d9 by the corresponding digit ei (1 ≤ i ≤ 9) is divisible by 7. The number f1 f2 . . f9 is related to e1 e2 . . e9 is the same way: that is, each of the nine numbers formed by replacing one of the ei by the corresponding fi is divisible by 7. Show that, for each i, di − fi is divisible by 7. [For example, if d1 d2 .

Let C1 and C2 be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find, with proof, the locus of all points M for which there exists points X on C1 and Y on C2 such that M is the midpoint of the line segment XY . (page 218) A3. Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses oﬀered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course. (page 218) A4.