18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2.

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1 18.S34 (FALL, 007) GREATEST INTEGER PROBLEMS Note: We use the otatio x for the greatest iteger x, eve if the origial source used the older otatio [x]. 1. (48P) If is a positive iteger, prove that = (a) Let p deote a prie uber, ad let be ay positive iteger. Show that the expoet of the highest power of p which divides! is + + +, p p p s where p s+1 >. (b) I how ay zeros does the uber 1000! ed, whe writte i base 10? 3. (a) Prove ( ) that the expoet of the highest power of p which divides is equal to the uber of carries that occur whe ad are added i base p (Kuer s theore). (b) For > 1 a coposite iteger, prove that ot all of ( ) ( ),..., 1 1 ca be divisible by. 4. Prove that for ay positive itegers i, j, k, is a iteger. (3i)!(3j)!(3k)! i!j!k!(i + j)!(j + k)!(k + i)! 5. Prove that for ay itegers 1,... k, the product j i j i is a iteger. 1 i<j k 1

2 6. (68IMO) For every atural uber, evaluate the su + k k = k+1 4 k+1 k=0 7. A sequece of real ubers is defied by the oliear first order recurrece u +1 = u (u 3). (a) If u 0 = 5/, give a siple forula for u. (b) If u 0 = 4, how ay digits (i base te) does u 10 have? 8. Defie a sequece a 1 < a < of positive itegers as follows. Pick a 1 = 1. Oce a 1,..., a have bee chose, let a +1 be the least positive iteger ot already chose ad ot of the for a i + i for 1 i. Thus a = is ot allowed, so a = 3. Now a + = 5 is also ot allowed, so a 3 = 4. The a = 7 is ot allowed, so a 4 = 6, etc. The sequece begis: 1, 3, 4, 6, 8, 9, 11, 1, 14, 16, 17, 19,.... Fid a siple forula for a. Your forula should eable you, for istace, to copute a 1,000, (a) (Proble A6, 93P; o cotestat solved it.) The ifiite sequece of s ad 3 s, 3, 3,, 3, 3, 3,, 3, 3, 3,, 3, 3,, 3, 3, 3,, 3, 3, 3,, 3, 3, 3,, 3, 3,, 3, 3, 3,,... has the property that, if oe fors a secod sequece that records the uber of 3 s betwee successive s, the result is idetical to the first sequece. Show that there exists a real uber r such that, for ay, the th ter of the sequece is if ad oly if = 1 + r for soe oegative iteger. (b) (siilar i flavor to (a), though ot ivolvig the greatest iteger fuctio) Let a 1, a,... be the sequece 1,,, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9,...

3 of itegers a defied as follows: a 1 = 1, a 1 a a 3, ad a is the uber of s appearig i the sequece. Fid real ubers α, c > 0 such that a li = c. α 10. (Proble B6, 95P; five of the top 04 cotestats received at least 9 poits (out of 10), ad o oe received 3 8 poits.) For a positive real uber α, defie S(α) = { α : = 1,, 3,...}. Prove that {1,, 3,...} caot be expressed as the disjoit uio of three sets S(α), S(β), ad S(γ). 11. Let be a positive iteger ad k ay iteger. Defie a sequece a, a +1,... as follows: a = k + a +1 = a,. Show that there exists a positive iteger N ad polyoials P 0 (),P 1 (),..., P N 1 () such that for all 0 i N 1 ad all itegers t for which tn + i, we have a tn+i = P i (t). 1. (Proble B1, 97P; 171 of the top 05 cotestats received 10 poits, ad 14 others received 8 9 poits.) Let {x} deote the distace betwee the real uber x ad the earest iteger. For each positive iteger, evaluate 6 1 ({ } { }) F = i,. 6 3 =1 (Here i(a, b) deotes the iiu of a ad b.) 13. (Proble B4, 98P; 73 of the top 199 cotestats received at least 8 poits.) Fid ecessary ad sufficiet coditios o positive itegers ad so that 1 ( 1) i/ + i/ = 0. i=0 3

4 14. (Proble B3, 01P; 9 of the top 00 cotestats received at least 8 poits.) For ay positive iteger, let deote the closest iteger to. Evaluate +. =1 15. (Proble B3, 03P; 15 of the top 01 cotestats received at least 8 poits.) Show that for each positive iteger,! = lc{1,,..., /i }. i=1 (Here lc deotes the least coo ultiple.) 16. Defie a 1 = 1 ad a +1 = a (a + 1), 1. Thus (a 1,..., a 10 ) = (1,, 3, 4, 6, 9, 13, 19, 7, 38). Show that a +1 a = 1, ad fid a siple descriptio of a +1 a Prove that for all positive itegers,, 1 k gcd(, ) = + +. k=0 18. Let a, b, c, d be real ubers such that a + b = c + d for all positive itegers. Prove that at least oe of a + b, a c, a d is a iteger. 19. Let p be a prie cogruet to 1 odulo 4. Prove that (p 1)/4 p 1 ip =. 1 i= Which positive itegers ca be writte i the for + + for soe positive iteger? 1. For a positive iteger, let x be the last digit i the decial represetatio of /. Is the sequece x 1, x,... periodic? 4

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