MathDB

1

New perfection of an integer [Baltic way 2003]

Suppose that the sum of all positive divisors of a natural number

n

,

n

excluded, plus the number of these divisors is equal to

n

. Prove that

n = 2m^2

for some integer

m

.

19

1

a^3+b^3 is a square [Baltic Way 2003]

Let

a

and

b

be positive integers. Show that if

a^3+b^3

is the square of an integer, then

a + b

is not a product of two different prime numbers.

18

1

All integers are coloured [Baltic way 2003]

Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if

a, b, c, d

are not all

0

and have the same colour, then

3a-2b \neq 2c-3d

?
[color=#0000FF][Mod edit: Question fixed]

17

1

Mary's claim to fame [Baltic Way 2003]

All the positive divisors of a positive integer

n

are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor

d > 1

whether it is a prime. Let

n

have

k

divisors not greater than

d

. Mary claims that it suffices to check divisibility of

d

by the first

\left\lceil\frac{k}{2}\right\rceil

divisors of

n

:

d

is prime if and only if none of them but

1

divides

d

. Is Mary right?

16

1

a-b is prime and ab is a square

Find all pairs of positive integers

(a,b)

such that

a-b

is a prime number and

ab

is a perfect square.

15

1

AT=RC [Baltic way 2003]

The diagonals of a cyclic convex quadrilateral

ABCD

intersect at

P

. A circle through

P

touches the side

CD

at its midpoint

M

and intersects the segments

BD

and

AC

again at the points

Q

and

R

respectively. Let

S

be the point on segment

BD

such that

BS = DQ

. The line through

S

parallel to

AB

intersects

AC

at

T

. Prove that

AT = RC

.

14

1

Concurrency from exterior equilateral triangles

Equilateral triangles

AMB,BNC,CKA

are constructed on the exterior of a triangle

ABC

. The perpendiculars from the midpoints of

MN, NK, KM

to the respective lines

CA, AB, BC

are constructed. Prove that these three perpendiculars pass through a single point.

12

1

Points M and N on the square ABCD

Points

M

and

N

are taken on the sides

BC

and

CD

respectively of a square

ABCD

so that

\angle MAN=45^{\circ}

. Prove that the circumcentre of

\triangle AMN

lies on

AC

.

11

1

1000 points in the plane [Baltic way 2003]

Is it possible to select

1000

points in the plane so that

6000

pairwise distances between them are equal?

10

1

Centroid of four points

A lattice point in the plane is a point with integral coordinates. The centroid of four points

(x_i,y_i )

,

i = 1, 2, 3, 4

, is the point

\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)

. Let

n

be the largest natural number for which there are

n

distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that

n = 12

.

9

1

10 questions

It is known that

n

is a positive integer and

n \le 144

. Ten questions of the type “Is

n

smaller than

a

?” are allowed. Answers are given with a delay: for

i = 1, \ldots , 9

, the

i

-th question is answered only after the

(i + 1)

-th question is asked. The answer to the tenth question is given immediately. Find a strategy for identifying

n

.

7

1

Maximise the cardinality of X

A subset of

X

of

\{1,2,3, \ldots 10000 \}

has the following property: If

a,b

are distinct elements of

X

, then

ab\not\in X

. What is the maximal number of elements in

X

?

6

1

Least number of indices with sum ≥ 0

Let

n\ge 2

and

d\ge 1

be integers with

d\mid n

, and let

x_1,x_2,\ldots x_n

be real numbers such that

x_1+x_2+\cdots + x_n=0

. Show that there are at least

\binom{n-1}{d-1}

choices of

d

indices

1\le i_1<i_2<\cdots <i_d\le n

such that

x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0

.

5

1

Sequence Inequality [Baltic Way 2003]

The sequence

(a_n)

is defined by

a_1=\sqrt{2}

,

a_2=2

, and

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

for

n\ge 2

. Prove that for every

n\ge 1

(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n.

4

1

a,b,c inequality [Baltic Way 2003]

Let

a,b,c

be positive real numbers. Prove that

\frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}

2

1

All real solutions satisfy inequality

Prove that any real solution of

x^3+px+q=0

, where

p,q

are real numbers, satisfies the inequality

4qx\le p^2

.

1

Function over the rationals

Find all functions

f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}

which for all

x \in \mathbb{Q}^{+}

fulfil

f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1).

8

1

problem with candy...

There are

2003

pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?

13

1

Reactangle

In a rectangle

ABCD

be a rectangle and

BC = 2AB

, let

E

be the midpoint of

BC

and

P

an arbitrary inner point of

AD

. Let

F

and

G

be the feet of perpendiculars drawn correspondingly from

A

to

BP

and from

D

to

CP

. Prove that the points

E,F,P,G

are concyclic.

3

1

Cyclic inequality with xyz=1

Let

x

,

y

and

z

be positive real numbers such that

xyz = 1

. Prove that

\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).

2003 Baltic Way

Subcontests

New perfection of an integer [Baltic way 2003]

a^3+b^3 is a square [Baltic Way 2003]

All integers are coloured [Baltic way 2003]

Mary's claim to fame [Baltic Way 2003]

a-b is prime and ab is a square

AT=RC [Baltic way 2003]

Concurrency from exterior equilateral triangles

Points M and N on the square ABCD

1000 points in the plane [Baltic way 2003]

Centroid of four points

10 questions

Maximise the cardinality of X

Least number of indices with sum ≥ 0

Sequence Inequality [Baltic Way 2003]

a,b,c inequality [Baltic Way 2003]

All real solutions satisfy inequality

Function over the rationals

problem with candy...

Reactangle

Cyclic inequality with xyz=1