MathDB

1

p + q is divisible by ...

p,q

be two consecutive odd prime numbers. Prove that

p+q

is a product of at least

3

natural numbers greater than

1

(not necessarily different).

20

1

p(p - c) = (p - a)(p - b) = S

Let

a\le b\le c

be the sides of a right triangle, and let

2p

be its perimeter. Show that p(p \minus{} c) \equal{} (p \minus{} a)(p \minus{} b) \equal{} S, where

S

is the area of the triangle.

19

1

Two circles inside another circle

Let

C

be a circle in plane. Let

C_1

and

C_2

be nonintersecting circles touching

C

internally at points

A

and

B

respectively. Let

t

be a common tangent of

C_1

and

C_2

touching them at points

D

and

E

respectively, such that both

C_1

and

C_2

are on the same side of

t

. Let

F

be the point of intersection of

AD

and

BE

. Show that

F

lies on

C

.

18

1

P>2D

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

17

1

Cyclic quadrilateral

Quadrangle

ABCD

is inscribed in a circle with radius 1 in such a way that the diagonal

AC

is a diameter of the circle, while the other diagonal

BD

is as long as

AB

. The diagonals intersect at

P

. It is known that the length of

PC

is

2/5

. How long is the side

CD

?

16

1

Polyhedron with parallelogram faces

All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

15

1

Noah and 8 animals

Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.

14

1

Connected towns

There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.

13

1

Inequality with 2n variables

Prove that for any positive

x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n

the inequality \sum_{i\equal{}1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i\equal{}1}^n(x_i\plus{}y_i)^2} holds.

12

1

Bijective integer function

Let

N

denote the set of natural numbers. Let

\phi: N\rightarrow N

be a bijective function and assume that there exists a finite limit \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L. What are the possible values of

L

?

11

1

Unique rational function

Let Q^\plus{} denote the set of positive rational numbers. Show that there exists one and only one function f: Q^\plus{}\to Q^\plus{} satisfying the following conditions: (i) If

0<q<1/2

then f(q)\equal{}1\plus{}f(q/(1\minus{}2q)), (ii) If

1<q\le2

then f(q)\equal{}1\plus{}f(q\minus{}1), (iii) f(q)\cdot f(1/q)\equal{}1 for all q\in Q^\plus{}.

10

1

Quartic polynomial

Find all fourth degree polynomial

p(x)

such that the following four conditions are satisfied: (i) p(x)\equal{}p(\minus{}x) for all

x

, (ii)

p(x)\ge0

for all

x

, (iii) p(0)\equal{}1 (iv)

p(x)

has exactly two local minimum points

x_1

and

x_2

such that |x_1\minus{}x_2|\equal{}2.

9

1

Cubic polynomial

A polynomial f(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}c is such that

b<0

and ab\equal{}9c. Prove that the polynomial

f

has three different real roots.

8

1

2^x (4-x) = 2x+4

Find all integers satisfying the equation 2^x\cdot(4\minus{}x)\equal{}2x\plus{}4.

7

1

Comparing two numbers

Let a\equal{}\sqrt[1992]{1992}. Which number is greater \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992} \text{or} 1992?

6

1

(k^3-1)/(k^3+1)

Prove that the product of the 99 numbers \frac{k^3\minus{}1}{k^3\plus{}1},k\equal{}2,3,\ldots,100 is greater than

2/3

.

2

1

d(n) divides n

Denote by

d(n)

the number of all positive divisors of a natural number

n

(including

1

and

n

). Prove that there are infinitely many

n

, such that

n/d(n)

is an integer.

5

1

^2 => ^4

It is given that a^2\plus{}b^2\plus{}(a\plus{}b)^2\equal{}c^2\plus{}d^2\plus{}(c\plus{}d)^2. Prove that a^4\plus{}b^4\plus{}(a\plus{}b)^4\equal{}c^4\plus{}d^4\plus{}(c\plus{}d)^4.

4

1

Squares of the lengths are consecutive integers

Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?

3

1

Arithmetic progression without sum of two squares/cubes

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

1992 Baltic Way

Subcontests

p + q is divisible by ...

p(p - c) = (p - a)(p - b) = S

Two circles inside another circle

P>2D

Cyclic quadrilateral

Polyhedron with parallelogram faces

Noah and 8 animals

Connected towns

Inequality with 2n variables

Bijective integer function

Unique rational function

Quartic polynomial

Cubic polynomial

2^x (4-x) = 2x+4

Comparing two numbers

(k^3-1)/(k^3+1)

d(n) divides n

^2 => ^4

Squares of the lengths are consecutive integers

Arithmetic progression without sum of two squares/cubes

1992 Baltic Way

Subcontests

p + q is divisible by ...

p(p - c) = (p - a)(p - b) = S

Two circles inside another circle

P&gt;2D

Cyclic quadrilateral

Polyhedron with parallelogram faces

Noah and 8 animals

Connected towns

Inequality with 2n variables

Bijective integer function

Unique rational function

Quartic polynomial

Cubic polynomial

2^x (4-x) = 2x+4

Comparing two numbers

(k^3-1)/(k^3+1)

d(n) divides n

^2 =&gt; ^4

Squares of the lengths are consecutive integers

Arithmetic progression without sum of two squares/cubes

P>2D

^2 => ^4