HMMT 二月 2000 · POW 赛 · 第 12 题

英文原题

Find the o eÆ ien t of X 2 Y 3 in ea h of the follo wing. (a) ( X + Y + 1 ) 7 (b) ( X 2 + Y 1 ) 7 13. Find the n um b er of "w ords" of length m from a set of n letters, if ea h letter m ust o ur at least on e in ea h w ord. 14. Find the n um b er of w a ys to distribute sev en distin t balls in to three distin t b o xes if ea h b o x m ust on tain a di eren t n um b er of balls, allo wing an empt y b o x. 15. Ho w man y w a ys an a lass of 10 studen ts b e divided in to t w o groups of 3 and 1 group of 4? 16. Find the n um b er of subsets A of the set of digits f 0 ; 1 ; 2 ; 3 ; : : : ; 9 g su h that A on tains no t w o onse utiv e digits. Hin t: Find a b etter statemen t of the problem; nd a re ursiv e form ula, and then attempt to solv e the problem for the n um b er of digits giv en. 17. If w e are trying to nd the n um b er of w ords of length m from a giv en set of n letters, with ea h letter o uring at least on e in ea h w ord, let us all the answ er T ( m; n ). This is equiv alen t to nding the n um b er of distribution of a set of m distin t balls in to a set of n distin t b o xes, if no b o xes an b e empt y . T ( m; n ) is the sum of all p ossible partitions of the balls (i.e. w e sum all p ossible w a ys of putting the balls in to b o xes (4 in b o x 1, 2 in b o x 2, 1 in b o x 3 for example)). More pre isely , if w e all m i to b e the n um b er of balls in b o x i , then T ( m; n ) = P m 1 + m 2 + m 3 = m m 1 ;m 2 ;m 3 > =1 m ! m 1 ! m 2 ! m 3 ! ::: m n ! . F or example, T (3 ; 2) = 3! 1!2! + 3! 2!1! = 3 + 3 = 6. Find a re ursiv e pattern for T ( m; n ) in terms of previous terms (previous meaning a smaller m , a smaller n , or b oth). Hin t: set up a sort of "P as al's T riangle" for T ( m; n ). Pro v e y our answ er using w ords. 18. Y ou ha v e an in nite n um b er of 1 en t, 2 en t, and 5 en t stamps. Y ou are trying to p ost a letter that requires n en ts of p ostage stamps, where n > 8. Let a ( n ) b e the n um b er of sequen es of stamps that giv e exa tly the required p ostage of n en ts (i.e. order matters). Find a ( n ) in terms of previous terms of the sequen e of a 's, using as few previous terms as p ossible. 19. Supp ose w e ha v e n lines in a plane in general p osition, whi h means that none are parallel to ea h other and that no three of these lines in terse t at a single p oin t. Find the n um b er of regions that these lines divide the plane in to... (a) in a re ursiv e form. (b) in a nonre ursiv e form ula. 20. Find the 2000th p ositiv e in teger that is not the di eren e b et w een an y t w o in teger squares.