MathDB

1

Sides and altitudes of are integers and perimeter is 1995

Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to

1995

?

14

1

Ienquality involving sides a,b,c and angles α, β, γ

Let

\alpha,\beta,\gamma

be the angles of a triangle opposite to its sides with lengths

a,b,c

respectively. Prove the inequality

a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)

12

1

O_1S_1, O_2S_2 and O_3S_3 are concurrent

The inscribed circle of the triangle

A_1A_2A_3

touches the sides

A_2A_3,A_3A_1,A_1A_2

at points

S_1,S_2,S_3

, respectively. Let

O_1,O_2,O_3

be the centres of the inscribed circles of triangles

A_1S_2S_3, A_2S_3S_1,A_3S_1S_2

, respectively. Prove that the straight lines

O_1S_1,O_2S_2,O_3S_3

intersect at one point.

11

1

|SA'| |SB'| = |SN|^2

Let

NS

and

EW

be two perpendicular diameters of a circle

\mathcal{C}

. A line

\ell

touches

\mathcal{C}

at point

S

. Let

A

and

B

be two points on

\mathcal{C}

, symmetric with respect to the diameter

EW

. Denote the intersection points of

\ell

with the lines

NA

and

NB

by

A'

and

B'

, respectively. Show that

|SA'|\cdot |SB'|=|SN|^2

.

10

1

Integer has 1994 digits from 1 to 5, next digit is 1 away

How many positive integers satisfy the following three conditions:
a) All digits of the number are from the set

\{1,2,3,4,5\}

; b) The absolute value of the difference between any two consecutive digits is

1

; c) The integer has

1994

digits?

9

1

2^a + 3^b is a square, classic diophantine equation

Find all pairs of positive integers

(a,b)

such that

2^a+3^b

is the square of an integer.

8

1

For any integer a, there exist b and c, where c^2=a^2+b^2

Show that for any integer

a\ge 5

there exist integers

b

and

c

,

c\ge b\ge a

, such that

a,b,c

are the lengths of the sides of a right-angled triangle.

7

1

Numerator is divisible by p

Let

p>2

be a prime number and

1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}

where

m

and

n

are relatively prime. Show that

m

is a multiple of

p

.

6

1

Beginner's problem, p/q can be expressed as n/(2^k-1)

Prove that any irreducible fraction

p/q

, where

p

and

q

are positive integers and

q

is odd, is equal to a fraction

\frac{n}{2^k-1}

for some positive integers

n

and

k

.

5

1

p(x)=1 and p(x)=3 have integer solutions, can p(x)=2 have 2?

Let

p(x)

be a polynomial with integer coefficients such that both equations

p(x)=1

and

p(x)=3

have integer solutions. Can the equation

p(x)=2

have two different integer solutions?

4

1

Some of two square roots is a rational number

Is there an integer

n

such that

\sqrt{n-1}+\sqrt{n+1}

is a rational number?

3

1

Begging for trig substitution

Find the largest value of the expression

xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}

2

1

Inequality involving 3 non-negative real numbers among 9

Let

a_1,a_2,\ldots ,a_9

be any non-negative numbers such that

a_1=a_9=0

and at least one of the numbers is non-zero. Prove that for some

i

,

2\le i\le 8

, the inequality

a_{i-1}+a_{i+1}<2a_i

holds. Will the statement remain true if we change the number

2

in the last inequality to

1.9

?

1

a∘b is defined as a+b-ab

Let

a\circ b=a+b-ab

. Find all triples

(x,y,z)

of integers such that

(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0

16

1

The wonder island

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them

120^{\circ}

. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

18

1

lines and numbers

There are

n>2

lines given in the plane. No two of the lines are parallel and no three of them intersect at one point. Every point of intersection of these lines is labelled with a natural number between

1

and

n-1

. Prove that, if and only if

n

is even, it is possible to assign the labels in such a way that every line has all the numbers from

1

to

n-1

at its points of intersection with the other

n-1

lines.

17

1

towns and islands

In a certain kingdom, the king has decided to build

25

new towns on

13

uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.

13

1

square contain 5 disks

Find the smallest number

a

such that a square of side

a

can contain five disks of radius

1

, so that no two of the disks have a common interior point.

20

1

Same colour triangle

An equilateral triangle is divided into

9000000

congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is coloured in one of three colours. Prove that there exist three points of the same colour being the vertices of a triangle with its sides parallel to the lines of the original triangle.

19

1

Spies

The Wonder Island Intelligence Service has

16

spies in Tartu. Each of them watches on some of his colleagues. It is known that if spy

A

watches on spy

B

, then

B

does not watch on

A

. Moreover, any

10

spies can numbered in such a way that the first spy watches on the second, the second watches on the third and so on until the tenth watches on the first. Prove that any

11

spies can also be numbered is a similar manner.

1994 Baltic Way

Subcontests

Sides and altitudes of are integers and perimeter is 1995

Ienquality involving sides a,b,c and angles α, β, γ

O_1S_1, O_2S_2 and O_3S_3 are concurrent

|SA'| |SB'| = |SN|^2

Integer has 1994 digits from 1 to 5, next digit is 1 away

2^a + 3^b is a square, classic diophantine equation

For any integer a, there exist b and c, where c^2=a^2+b^2

Numerator is divisible by p

Beginner's problem, p/q can be expressed as n/(2^k-1)

p(x)=1 and p(x)=3 have integer solutions, can p(x)=2 have 2?

Some of two square roots is a rational number

Begging for trig substitution

Inequality involving 3 non-negative real numbers among 9

a∘b is defined as a+b-ab

The wonder island

lines and numbers

towns and islands

square contain 5 disks

Same colour triangle

Spies