173 bài toán về dãy số trong các kỳ thi olympic quốc tế

SEQUENCE

1. The sequence an is defined as follows: a1 = 1, an+1 = an + 1/an for n ≥ 1. Prove that a100

> 14. (ASU 1968)

2. 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)

3. 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)

4. a1 and a2 are positive integers less than 1000. Define an = min{|ai - aj| : 0 < i < j<n}.

Show that a21=0. (ASU 1976)

5. an is an infinite sequence such that (an+1 - an)/2 tends to zero. Show that an tends to

zero.(ASU1977)

6. 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)

7. 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)

8. 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)

9. 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)

10.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)

11.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)

12.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 ∑

=

n

i

i

a

1

.(ASU 1982)

13.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)

14.The real sequence xn is defined by x1 = 1, x2 = 1, xn+2 = xn+1

2

- xn/2. Show that the

sequence converges and find the limit.(ASU 1984)

15.The sequence a1, a2, a3, ... satisfies a4n+1 = 1, a4n+3 = 0, a2n = an. Show that it is not

periodic.(ASU 1985)

1