SEQUENCE
The sequence an is defined as follows: a1 = 1, an+1 = an + 1/an for n
1. Prove that a100 > 14. (ASU 1968)
The sequence a1, a2, ... , an satisfies the following conditions: a1 = 0, |ai| = |ai-1 + 1| for i = 2, 3, ... , n. Prove that (a1 + a2 + ... + an)/n
-1/2. (ASU 1968)
A sequence of finite sets of positive integers is defined as follows. S0 = {m}, where m > 1. Then given Sn you derive Sn+1 by taking k2 and k+1 for each element k of Sn. For example, if S0 = {5}, then S2 = {7, 26, 36, 625}. Show that Sn always has 2n distinct elements.(ASU 1972)
a1 and a2 are positive integers less than 1000. Define an = min{|ai - aj| : 0 < i < jan is an infinite sequence such that (an+1 - an)/2 tends to zero. Show that an tends to zero.(ASU1977)
Given a sequence a1, a2, ... , an of positive integers. Let S be the set of all sums of one or more members of the sequence. Show that S can be divided into n subsets such that the smallest member of each subset is at least half the largest member.(ASU 1977)
Show that there is an infinite sequence of reals x1, x2, x3, ... such that |xn| is bounded and for any m > n, we have |xm - xn| > 1/(m - n).(ASU 1978)
The real sequence x1
x2
x3
... satisfies x1 + x4/2 + x9/3 + x16/4 + ... + xN/n
1 for every square N = n2. Show that it also satisfies x1 + x2/2 + x3 /3 + ... + xn/n
3. (ASU1979)
Define the sequence an of positive integers as follows. a1 = m. an+1 = an plus the product of the digits of an. For example, if m = 5, we have 5, 10, 10, ... . Is there an m for which the sequence is unbounded?(ASU 1980)
The sequence an of positive integers is such that (1) an
n3/2 for all n, and (2) m-n divides km - kn (for all m > n). Find an.(ASU 1981)
The sequence an is defined by a1 = 1, a2 = 2, an+2 = an+1 + an. The sequence bn is defined by b1 = 2, b2 = 1, bn+2 = bn+1 + bn. How many integers belong to both sequences?(ASU1982)
A subsequence of the sequence real sequence a1, a2, ... , an is chosen so that (1) for each i at least one and at most two of ai, ai+1, ai+2 are chosen and (2) the sum of the absolute values of the numbers in the subsequence is at least 1/6
.(ASU 1982)
an is the last digit of [10n/2]. Is the sequence an periodic? bn is the last digit of [2n/2]. Is the sequence bn periodic?(ASU 1983)
The real sequence xn is defined by x1 = 1, x2 = 1, xn+2 = xn+12 - xn/2. Show that the sequence converges and find the limit.(ASU 1984)
The sequence a1, a2, a3, ... satisfies a4n+1 = 1, a4n+3 = 0, a2n = an. Show that it is not periodic.(ASU 1985)
The sequence of integers an is given by a0 = 0, an = p(an-1), where p(x) is a polynomial whose coefficients are all positive integers. Show that for any two positive integers m, k with greatest common divisor d, the greatest common divisor of am and ak is ad.(ASU 1988)
A sequence of positive integers is constructed as follows. If the last digit of an is greater than 5, then an+1 is 9an. If the last digit of an is 5 or less and an has more than one digit, then an+1 is obtained from an by deleting the last digit. If an has only one digit, which is 5 or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?(ASU 1991)
Define the sequence a1 = 1, a2, a3, ... by an+1 = a12 + a2 2 + a32 + ... + an2 + n. Show that 1 is the only square in the sequence. (CIS 1992)
The sequence (an) satisfies am+n+ am-n=
(a2m+a2n) for all m
n
0. If a1=1, find a1995. (Russian 1995)
The sequence a1, a2, a3, ... of positive integers is determined by its first two members and the rule an+2 = (an+1 + an)/gcd(an, an+1). For which values of a1 and a2 is it bounded?(Russian 1999)
The sequence a1, a2, .... , a3972 includes each of the numbers from 1 to 1986 twice. Can the terms be rearranged so that there are just n numbers between the two n`s?(CMO 1986)
The integer sequence ai is defined by a0 = m, a1 = n, a2 = 2n-m+2, ai+3 =3(ai+2 - ai+1) + ai. It contains arbitrarily long sequences consecutive terms which are squares. Show that every term is a square.(CMO 1992)
x0, x1, ... , is a sequence of binary strings of length n. n is odd and x0 = 100...01. xm+1 is derived from xm as follows: the kth digit in the string is 0 if the kth and k+1st digits in the previous string are the same, 1 otherwise. [The n+1th digit in a string means the 1st]. Show that if xm = xn, then m is a multiple of n].(CMO 1995)
a1, a2, ... is a sequence of non-negative integers such that an+m
an + am for all m, n. Show that if N
n, then an + aN
na1 + N/n an.(CMO 1997)
The sequence an is defined by a1 = 0, a2 = 1, an = (n an-1 + n(n-1) an-2 + (-1)n-1n)/2 + (-1)n. Find an + 2 nC1 an-1 + 3 nC2 an-2 + ... + n nC(n-1) a1, where nCm is the binomial coefficient n!/(m! (n-m)! ).(CMO 2000)
Let a1 = 0, a2n+1 = a2n = n. Let s(n) = a1 + a2 + ... + an. Find a formula for s(n) and show that s(m + n) = mn + s(m - n) for m > n.(CanMO 1970)
Let an = 1/(n(n+1) ). (1) Show that 1/n = 1/(n+1) + an. (2) Show that for any integer n > 1 there are positive integers r < s such that 1/n = ar + ar+1 + ... + as.(CanMO 1973)
Define the real sequence a1, a2, a3, ... by a1 = 1/2, n2an = a1 + a2 + ... + an. Evaluate an. (CanMO 1975)
The real sequence x0, x1, x2, ... is defined by x0 = 1, x1 = 2, n(n+1) xn+1 = n(n-1) xn - (n-2) xn-1. Find x0/x1 + x1x2 + ... + x50/x51.(CanMO 1976)
The real sequence x1, x2, x3, ... is defined by x1 = 1 + k, xn+1 = 1/xn + k, where 0 < k < 1. Show that every term exceeds 1.(CanMO 1977)
Define the real sequence x1, x2, x3, ... by x1 = k, where 1 < k < 2, and xn+1 = xn - xn2/2 + 1. Show that |xn -
| < 1/2n for n > 2.(CanMO 1985)
The integer sequence a1, a2, a3, ... is defined by a1 = 39, a2 = 45, an+2 = an+12 - an. Show that infinitely many terms of the sequence are divisible by 1986.(CanMO 1986)
Define two integer sequences a0, a1, a2, ... and b0, b1, b2, ... as follows. a0 = 0, a1= 1, an+2 = 4an+1 - an, b0 = 1, b1 = 2, bn+2 = 4bn+1 - bn. Show that bn2 = 3an2 + 1.(CanMO 1988)
A sequence of positive integers a1, a2, a3, ... is defined as follows. a1 = 1, a2 = 3, a3 = 2, a4n = 2a2n, a4n+1 = 2a2n + 1, a4n+2 = 2a2n+1 + 1, a4n+3 = 2a2n+1. Show that the sequence is a permutation of the positive integers. (CanMO 1993)
Show that non-negative integers a
b satisfy (a2 + b2) = n2(ab + 1), where n is a positive integer, if they are consecutive terms in the sequence ak defined by a0 = 0, a1 = n, ak+1 = n2ak - ak-1. (CanMO 1998)
Show that in any sequence of 2000 integers each with absolute value not exceeding 1000 such that the sequence has sum 1, we can find a subsequence of one or more terms with zero sum.(CanMO 2000)
Each member of the sequence a1, a2, ... , an belongs to the set {1, 2, ... , n-1} and a1 + a2 + ... + an < 2n. Show that we can find a subsequence with sum n.(Irish 1988)
The sequence of nonzero reals x1, x2, x3, ... satisfies xn = xn-2xn-1/(2xn-2 - xn-1) for all n > 2. For which (x1, x2) does the sequence contain infinitely many integral terms?(Irish 1988)
The sequence a1, a2, a3, ... is defined by a1 = 1, a2n = an, a2n+1 = a2n + 1. Find the largest value in a1, a2, ... , a1989 and the number of times it occurs.(Irish 1989)
The sequence a1, a2, a3, ... is defined by a1 = 1, a2n = an, a2n+1 = a2n + 1. Find the largest value in a1, a2, ... , a1989 and the number of times it occurs.(Irish 2002)
The sequence
is defined as: x1=1, xn+1=x n2- 3xn + 4,n= 1,2,3,... a) Prove that
is monotone increasing and unbounded. b) Prove that the sequence
defined as yn = 1/(x1-1) +....+1/(xn-1) is convergent and find its limit (Bungari 1997-Problem in winter)
Let
be a sequence of integer number such that their dicemal representations consist of even digits( a1=2, a2=4, a3=6,...). Find all integer number m such that am= 12m.(Bungari 1998 - Problem in winter)
Prove that for every positive number
the sequence
such that x1=1, x2=a,
=
,n
1 is convergent and find its limit.(Bungari 2000-Problem11.1)
Given the sequence
=

, n=1,2,.....where
is a real number: a) Find the values of a such that the sequence
is convergent. b) Find the values of a such that the sequence
is monotone increasing.(Bungari 1999-Pro in winter)
Let
be a sequence such that x1=43, x2=142,
= 3
+
,n
.Prove that: a)
and
are relatively prime for all n. b) for every natural number m there exits infinitely many natural number n such that
-1 and
-1 both divisible by m. (Bungari 2000-Pro3 third round)
A sequence is a1, a2, a3,.... is defined by a1= k, a2= 5k-2 and an+2= 3an+1- 2an, n
1, where k is a real number a)Find all values of k, such that the sequence
is convergent. b)Prove that if k=1 then:
,n
1, where
denoted the integer part of x.(Bungari 2001,2-4)
Define the sequence a1, a2, a3, ... by a1 = 1, an = an-1 - n if an-1 > n, an-1 + n if an-1
n. Let S be the set of n such that an = 1993. Show that S is infinite. Find the smallest member of S. If the element of S are written in ascending order show that the ratio of consecutive terms tends to 3.(IMO SHORTLIST 1993)
The sequence x0, x1, x2, ... is defined by x0 = 1994, xn+1 = xn2/(xn + 1). Show that [xn ] = 1994 - n for 0
n
998.(IMO SHORTLIST 1994)
Define the sequences an, bn, cn as follows. a0 = k, b0 = 4, c0 = 1. If an is even then an+1 = an/2, bn+1 = 2bn, cn+1 = cn. If an is odd, then an+1 = an - bn/2 - cn, bn+1 = bn, cn+1 = bn + cn. Find the number of positive integers k < 1995 such that some an = 0.