MathDB

1

(a_2+a_3+...a_n)/a_1 x a_1+a_3+a_4+...a_n)/a_2...

n\ge 3

be a given integer. Nonzero real numbers

a_1,..., a_n

satisfy:

\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}

What values can be taken by the product

\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}

?

9

1

consider words composed of n letters A and n letters $B

Given an integer

n > 1

, consider words composed of

n

letters

A

and

n

letters

B

. A word

X_1...X_{2n}

is said to belong to set

R(n)

(respectively,

S(n)

) if no initial segment (respectively, exactly one initial segment)

X_1...X_k

with

1 \le k < 2n

consists of equally many letters

A

and

B

. If

r(n)

and

s(n)

denote the cardinalities of

R(n)

and

S(n)

respectively, compute

s(n)/r(n)

.

6

1

f (92+ x) = f (92- x),f (19 .92+x) = f(19. 92-x),f (1992 + x) = f (1992 -x)

A function

f: Z \to Z

has the following properties:

f (92 + x) = f (92 - x)

f (19 \cdot 92 + x) = f (19 \cdot 92 - x)

(

19 \cdot 92 = 1748

)

f (1992 + x) = f (1992 - x)

for all integers

x

. Can all positive divisors of

92

occur as values of f?

4

1

P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c

Let

k

be a positive integer and

u, v

be real numbers. Consider

P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c

. (a) For

k = 2

prove that if

a, b, c

are rational then so is

uv

. (b) Is that also true for

k = 3

?

3

1

2\sqrt{bc + ca + ab} <= \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}

For all positive numbers

a, b, c

prove the inequality

2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}

.

2

1

each point on the boundary of a square has to be colored in one color

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

1

s(n - 1)s(n)s(n + 1) is even where s(n) is sum of all positive divisors of n

For a natural number

n

, denote by

s(n)

the sum of all positive divisors of n. Prove that for every

n > 1

the product

s(n - 1)s(n)s(n + 1)

is even.

7

1

< APB, < BPC, < CPA right angles in space, given triangle ABC

Consider triangles

ABC

in space. (a) What condition must the angles

\angle A, \angle B , \angle C

of

\triangle ABC

fulfill in order that there is a point

P

in space such that

\angle APB, \angle BPC, \angle CPA

are right angles? (b) Let

d

be the longest of the edges

PA,PB,PC

and let

h

be the longest altitude of

\triangle ABC

. Show that

\frac{1}{3}\sqrt6 h \le d \le h

.

5

1

fixed circle, fixed diameter, fixed point lead to constant product AP\cdot AQ

Given a circle

k

with center

M

and radius

r

, let

AB

be a fixed diameter of

k

and let

K

be a fixed point on the segment

AM

. Denote by

t

the tangent of

k

at

A

. For any chord

CD

through

K

other than

AB

, denote by

P

and Q the intersection points of

BC

and

BD

with

t

, respectively. Prove that

AP\cdot AQ

does not depend on

CD

.

1992 Austrian-Polish Competition

Subcontests

(a_2+a_3+...a_n)/a_1 x a_1+a_3+a_4+...a_n)/a_2...

consider words composed of n letters A and n letters $B

f (92+ x) = f (92- x),f (19 .92+x) = f(19. 92-x),f (1992 + x) = f (1992 -x)

P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c

2\sqrt{bc + ca + ab} <= \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}

each point on the boundary of a square has to be colored in one color

s(n - 1)s(n)s(n + 1) is even where s(n) is sum of all positive divisors of n

< APB, < BPC, < CPA right angles in space, given triangle ABC

fixed circle, fixed diameter, fixed point lead to constant product AP\cdot AQ

1992 Austrian-Polish Competition

Subcontests

(a_2+a_3+...a_n)/a_1 x a_1+a_3+a_4+...a_n)/a_2...

consider words composed of n letters A and n letters $B

f (92+ x) = f (92- x),f (19 .92+x) = f(19. 92-x),f (1992 + x) = f (1992 -x)

P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c

2\sqrt{bc + ca + ab} &lt;= \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}

each point on the boundary of a square has to be colored in one color

s(n - 1)s(n)s(n + 1) is even where s(n) is sum of all positive divisors of n

&lt; APB, &lt; BPC, &lt; CPA right angles in space, given triangle ABC

fixed circle, fixed diameter, fixed point lead to constant product AP\cdot AQ

2\sqrt{bc + ca + ab} <= \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}

< APB, < BPC, < CPA right angles in space, given triangle ABC