PEN M Problems

Part of PEN Problems

Subcontests

(35)

1

M 35

The first four terms of an infinite sequence

S

of decimal digits are

1

9

8

2

, and succeeding terms are given by the final digit in the sum of the four immediately preceding terms. Thus

S

1

9

8

2

0

9

9

0

8

6

3

7

4

\cdots

. Do the digits

3

0

4

4

ever come up consecutively in

S

1

M 34

The sequence of integers

\{ x_{n}\}_{n\ge1}

is defined as follows:

x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).

Prove that there are no squares of natural numbers in this sequence except

x_{1}

1

M 33

\{x_{n}\}_{n \ge 1}

is defined by x_{1} \equal{} 2, x_{n \plus{} 1} \equal{} \frac {2 \plus{} x_{n}}{1 \minus{} 2x_{n}}\;\; (n \in \mathbb{N}). Prove that a) x_{n}\not \equal{} 0 for all

n \in \mathbb{N}

\{x_{n}\}_{n \ge 1}

is not periodic.

1

M 32

In an increasing infinite sequence of positive integers, every term starting from the

2002

-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

1

M 31

Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.

1

M 30

k

be a positive integer. Prove that there exists an infinite monotone increasing sequence of integers

\{a_{n}\}_{n \ge 1}

a_{n}\; \text{divides}\; a_{n+1}^{2}+k \;\; \text{and}\;\; a_{n+1}\; \text{divides}\; a_{n}^{2}+k

n \in \mathbb{N}

1

M 29

\{a_{n}\}_{n \ge 1}

a_{1}=1

a_{n+1}= \frac{a_{n}}{2}+\frac{1}{4a_{n}}\; (n \in \mathbb{N}).

\sqrt{\frac{2}{2a_{n}^{2}-1}}

is a positive integer for

n>1

1

M 28

\{u_{n}\}_{n \ge 0}

be a sequence of integers satisfying the recurrence relation

u_{n+2}=u_{n+1}^2 -u_{n}

(n \in \mathbb{N})

u_{0}=39

u_{1}=45

1986

divides infinitely many terms of this sequence.

1

M 27

p \ge 3

be a prime number. The sequence

\{a_{n}\}_{n \ge 0}

a_{n}=n

0 \le n \le p-1

a_{n}=a_{n-1}+a_{n-p}

n \ge p

a_{p^{3}}\; \pmod{p}

1

M 26

p

be an odd prime

p

2h \neq 1 \; \pmod{p}

h \in \mathbb{N}

h< p-1

a

be an even integer with

a \in] \tfrac{p}{2}, p [

\{a_n\}_{n \ge 0}

a_{0}=a

a_{n+1}=p -b_{n}

(n \ge 0)

b_{n}

is the greatest odd divisor of

a_n

. Show that the sequence

\{a_n\}_{n \ge 0}

is periodic and find its minimal (positive) period.

1

M 25

\{a_{n}\}_{n \ge 1}

be a sequence of positive integers such that

0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.

Show that there are infinitely many pairs

(p, q)

of positive integers such that

p>q

a_{q}\; \vert \; a_{p}

1

M 24

k

be a given positive integer. The sequence

x_n

is defined as follows:

x_1 =1

x_{n+1}

is the least positive integer which is not in

\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}

. Show that there exist real number

a

x_n = \lfloor an\rfloor

for all positive integer

n

1

M 23

\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}

d(n, m)

are integers for all

m, n \in \mathbb{N}

1

M 22

\, a

b \,

be odd positive integers. Define the sequence

\{f_n\}_{n\ge 1}

\, f_1 = a,

f_2 = b, \,

\, f_n \,

\, n \geq 3 \,

be the greatest odd divisor of

\, f_{n-1} + f_{n-2}

\, f_n \,

is constant for sufficiently large

\, n \,

and determine the eventual value as a function of

\, a \,

\, b

1

M 21

In the sequence

1, 0, 1, 0, 1, 0, 3, 5, \cdots

, each member after the sixth one is equal to the last digit of the sum of the six members just preceeding it. Prove that in this sequence one cannot find the following group of six consecutive members:

0, 1, 0, 1, 0, 1

1

M 20

Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

1

M 19

A sequence with first two terms equal

1

24

respectively is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.

1

M 18

Given is an integer sequence

\{a_n\}_{n \ge 0}

a_{0}=2

a_{1}=3

and, for all positive integers

n \ge 1

a_{n+1}=2a_{n-1}

a_{n+1}= 3a_{n} - 2a_{n-1}

. Does there exist a positive integer

k

1600 < a_{k} < 2000

1

M 17

A sequence of integers,

\{a_{n}\}_{n \ge 1}

a_{1}>0

, is defined by

a_{n+1}=\frac{a_{n}}{2}\;\;\; \text{if}\;\; n \equiv 0 \;\; \pmod{4},

a_{n+1}=3 a_{n}+1 \;\;\; \text{if}\;\; n \equiv 1 \; \pmod{4},

a_{n+1}=2 a_{n}-1 \;\;\; \text{if}\;\; n \equiv 2 \; \pmod{4},

a_{n+1}=\frac{a_{n}+1}{4}\;\;\; \text{if}\;\; n \equiv 3 \; \pmod{4}.

Prove that there is an integer

m

a_{m}=1

1

M 16

Define a sequence

\{a_i\}

a_1=3

a_{i+1}=3^{a_i}

i\geq 1

. Which integers between

00

99

inclusive occur as the last two digits in the decimal expansion of infinitely many

a_i

1

M 15

For a given positive integer

k

denote the square of the sum of its digits by

f_{1}(k)

f_{n+1}(k)=f_{1}(f_{n}(k))

. Determine the value of

f_{1991}(2^{1990})

1

M 14

x_{1}

x_{2}

be relatively prime positive integers. For

n \ge 2

x_{n+1}=x_{n}x_{n-1}+1

.[*] Prove that for every

i>1

j>i

{x_{i}}^{i}

{x_{j}}^{j}

. [*] Is it true that

x_{1}

{x_{j}}^{j}

j>1

1

M 13

\{x_{n}\}

x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.

Prove that the sequence is periodic if and only if

x_{0}

1

M 12

k

be a fixed positive integer. The sequence

\{a_{n}\}_{n\ge1}

a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.

m \neq n

, then the numbers

a_{m}

a_{n}

are relatively prime.

1

M 11

a_{1}={11}^{11}

a_{2}={12}^{12}

a_{3}={13}^{13}

a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.

a_{{14}^{14}}

1

M 10

An integer sequence satisfies

a_{n+1}={a_n}^3 +1999

. Show that it contains at most one square.

1

M 9

An integer sequence

\{a_{n}\}_{n \ge 1}

a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.

Show that it has infinitely many even and infinitely many odd integers.

1

M 8

The Bernoulli sequence

\{B_{n}\}_{n \ge 0}

B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)

Show that for all

n \in \mathbb{N}

(-1)^{n}B_{n}-\sum \frac{1}{p},

is an integer where the summation is done over all primes

p

p| 2k-1

1

M 7

Prove that the sequence

\{y_{n}\}_{n \ge 1}

y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right)

consists only of integers.

1

M 6

\{a_{n}\}_{n \ge 1}

a_{1}=1, \; a_{n+1}=2a_{n}+\sqrt{3a_{n}^{2}+1}.

a_{n}

is an integer for every

n

1

M 5

Show that there is a unique sequence of integers

\{a_{n}\}_{n \ge 1}

a_{1}=1, \; a_{2}=2, \; a_{4}=12, \; a_{n+1}a_{n-1}=a_{n}^{2}\pm1 \;\; (n \ge 2).

1

M 4

\{a_{n}\}_{n \ge 1}

a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).

a_{n}

is an integer for all

n

, and show that

n|a_{n}

n\in\mathbb{N}

1

M 3

f(n)=n+\lfloor \sqrt{n}\rfloor

. Prove that, for every positive integer

m

m, f(m), f(f(m)), f(f(f(m))), \cdots

contains at least one square of an integer.

1

M 2

An integer sequence

\{a_{n}\}_{n \ge 1}

a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.

a_{n}

is a square if and only if

n=2^{k}+k-2

k \in \mathbb{N}

1

M 1

P(x)

be a nonzero polynomial with integer coefficients. Let

a_{0}=0

i \ge 0

a_{i+1}=P(a_{i})

\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}

m, n \in \mathbb{N}