MathDB

1

Obtaining sequences through set steps of (a,b,c,d)

(a, b, c, d)

each of the sequences (c, d, a, b), (b, a, d, c), (a + nc, b + nd, c, d), (a + nb, b, c + nd, d) for arbitrary integer

n

can be obtained by one step. Is it possible to obtain

(3, 4, 5, 7)

from

(1, 2, 3, 4)

through a sequence of such steps?

19

1

Same number of factors as 360

What is the smallest positive odd integer having the same number of positive divisors as

360

?

17

1

2^{n-1}+n numbers can be chosed from the set

Let

n

be a positive integer. Prove that at least

2^{n-1}+n

numbers can be chosen from the set

\{1, 2, 3,\ldots ,2^n\}

such that for any two different chosen numbers

x

and

y

,

x+y

is not a divisor of

x\cdot y

.

16

1

Find f(2002)

Let

f

be a real-valued function defined on the positive integers satisfying the following condition: For all

n>1

there exists a prime divisor

p

of

n

such that

f(n)=f\left(\frac{n}{p}\right)-f(p)

. Given that

f(2001)=1

, what is the value of

f(2002)

?

15

1

Sequence inequality

Let

a_0,a_1,a_2,\ldots

be a sequence of positive real numbers satisfying

i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}

for

i=1, 2, \ldots

Furthermore, let

x

and

y

be positive reals, and let

b_i=xa_i+ya_{i-1}

for

i=1, 2, \ldots

Prove that the inequality

i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}

holds for all integers

i\ge 2

.

14

1

2n cards

There are

2n

cards. On each card some real number

x

,

(1\le x\le 2n)

, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums

s_1

and

s_2

so that

\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1

.

11

1

Function acting on a,b, and gcd(a,b)

The real-valued function

f

is defined for all positive integers. For any integers

a>1, b>1

with

d=\gcd (a, b)

, we have

f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right)

Determine all possible values of

f(2001)

.

13

1

There exists k such that 2001!a_k<k

Let

a_0, a_1, a_2,\ldots

be a sequence of real numbers satisfying

a_0=1

and

a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}

for

n=1, 2,\ldots

Prove that there exists a positive integer

k

with

a_k<\frac{k}{2001!}

.

12

1

Sum of third powers equals 3, sum of fifth powers equal 5

Let

a_1, a_2,\ldots , a_n

be positive real numbers such that

\sum_{i=1}^na_i^3=3

and

\sum_{i=1}^na_i^5=5

. Prove that

\sum_{i=1}^na_i>\frac{3}{2}

.

10

1

Determine the angles of ABC

In a triangle

ABC

, the bisector of

\angle BAC

meets the side

BC

at the point

D

. Knowing that

|BD|\cdot |CD|=|AD|^2

and

\angle ADB=45^{\circ}

, determine the angles of triangle

ABC

.

9

1

The locus of P with supplementary angles condition

Given a rhombus

ABCD

, find the locus of the points

P

lying inside the rhombus and satisfying

\angle APD+\angle BPC=180^{\circ}

.

8

1

Inequality in convex quadrilateral ABCD

Let

ABCD

be a convex quadrilateral, and let

N

be the midpoint of

BC

. Suppose further that

\angle AND=135^{\circ}

. Prove that

|AB|+|CD|+\frac{1}{\sqrt{2}}\cdot |BC|\ge |AD|.

7

1

Circle through A of parallelogram ABCD

Given a parallelogram

ABCD

. A circle passing through

A

meets the line segments

AB, AC

and

AD

at inner points

M,K,N

, respectively. Prove that

|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|

6

1

5 concyclic points A,B,C,D,E have AB||EC, AC||ED

The points

A, B, C, D, E

lie on the circle

c

in this order and satisfy

AB\parallel EC

and

AC\parallel ED

. The line tangent to the circle

c

at

E

meets the line

AB

at

P

. The lines

BD

and

EC

meet at

Q

. Prove that

|AC|=|PQ|

.

5

1

Colouring of 2001 points on a circle

Let

2001

given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point

P

have the same colour as

P

, then the colour of

P

remains unchanged, otherwise

P

obtains the other colour. Starting with the first colouring

F_1

, we obtain the colourings

F_2,F_3 ,\ldots .

after several recolouring steps. Prove that there is a number

n_0\le 1000

such that

F_{n_0}=F_{n_0 +2}

. Is the assertion also true if

1000

is replaced by

999

?

4

1

Sum of floors with primes p,q

Let

p

and

q

be two different primes. Prove that

\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1)

3

1

Numbers placed in a 7x7 array

The numbers

1, 2, \ldots 49

are placed in a

7\times 7

array, and the sum of the numbers in each row and in each column is computed. Some of these

14

sums are odd while others are even. Let

A

denote the sum of all the odd sums and

B

the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that

A = B

?

2

1

Each expressible as n integers from different sets

Let

n\ge 2

be a positive integer. Find whether there exist

n

pairwise nonintersecting nonempty subsets of

\{1, 2, 3, \ldots \}

such that each positive integer can be expressed in a unique way as a sum of at most

n

integers, all from different subsets.

1

Largest number of students in an 8-problem exam

A set of

8

problems was prepared for an examination. Each student was given

3

of them. No two students received more than one common problem. What is the largest possible number of students?

18

1

Relatively prime proof

Let

a

be an odd integer. Prove that

a^{2^m}+2^{2^m}

and

a^{2^n}+2^{2^n}

are relatively prime for all positive integers

n

and

m

with

n\not= m

.

2001 Baltic Way

Subcontests

Obtaining sequences through set steps of (a,b,c,d)

Same number of factors as 360

2^{n-1}+n numbers can be chosed from the set

Find f(2002)

Sequence inequality

2n cards

Function acting on a,b, and gcd(a,b)

There exists k such that 2001!a_k<k

Sum of third powers equals 3, sum of fifth powers equal 5

Determine the angles of ABC

The locus of P with supplementary angles condition

Inequality in convex quadrilateral ABCD

Circle through A of parallelogram ABCD

5 concyclic points A,B,C,D,E have AB||EC, AC||ED

Colouring of 2001 points on a circle

Sum of floors with primes p,q

Numbers placed in a 7x7 array

Each expressible as n integers from different sets

Largest number of students in an 8-problem exam

Relatively prime proof

2001 Baltic Way

Subcontests

Obtaining sequences through set steps of (a,b,c,d)

Same number of factors as 360

2^{n-1}+n numbers can be chosed from the set

Find f(2002)

Sequence inequality

2n cards

Function acting on a,b, and gcd(a,b)

There exists k such that 2001!a_k&lt;k

Sum of third powers equals 3, sum of fifth powers equal 5

Determine the angles of ABC

The locus of P with supplementary angles condition

Inequality in convex quadrilateral ABCD

Circle through A of parallelogram ABCD

5 concyclic points A,B,C,D,E have AB||EC, AC||ED

Colouring of 2001 points on a circle

Sum of floors with primes p,q

Numbers placed in a 7x7 array

Each expressible as n integers from different sets

Largest number of students in an 8-problem exam

Relatively prime proof

There exists k such that 2001!a_k<k