MathDB

1

max of f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}

n

determine the maximum value of the function

f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}

over all

x \ge 0

and find all positive

x

for which the maximum is attained.

9

1

f: A \to A, f(k)\ne g(k) and f(f(f(k))= g(k)$ for k \in A

For a positive integer

n

denote

A = \{1,2,..., n\}

. Suppose that

g : A\to A

is a fixed function with

g(k) \ne k

and

g(g(k)) = k

for

k \in A

. How many functions

f: A \to A

are there such that

f(k)\ne g(k)

and

f(f(f(k))= g(k)

for

k \in A

?

8

1

xy=- 1 (mod z), yz = 1 (mod x), zx = 1 (mod y).

Consider the system of congruences

\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}

Find the number of triples

(x,y, z)

of distinct positive integers satisfying this system such that one of the numbers

x,y, z

equals

19

.

5

1

x^2+y^2+z^2 + xy+yz + zx> =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z>0

If

x,y, z

are arbitrary positive numbers with

xyz = 1

, prove the inequality

x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})

.

4

1

P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)

Let

P(x)

be a real polynomial with

P(x) \ge 0

for

0 \le x \le 1

. Show that there exist polynomials

P_i (x) (i = 0, 1,2)

with

P_i (x) \ge 0

for all real x such that

P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)

.

1

(m \choose 2) = 3 (n \choose 4) combinations

Show that there are infinitely many integers

m \ge 2

such that

m \choose 2

= 3

n \choose 4

holds for some integer

n \ge 4

. Give the general form of all such

m

.

3

1

P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and midpoints A_1, A_2, A_3, A_1,..

Given two distinct points

A_1,A_2

in the plane, determine all possible positions of a point

A_3

with the following property: There exists an array of (not necessarily distinct) points

P_1,P_2,...,P_n

for some

n \ge 3

such that the segments

P_1P_2,P_2P_3,...,P_nP_1

have equal lengths and their midpoints are

A_1, A_2, A_3, A_1, A_2, A_3, ...

in this order.

2

1

(x^2 - 6x + 13)y = 20,(y^2 - 6y + 13)z = 20, (z^2 - 6z + 13)x = 20

Find all solutions

(x,y,z)

to the system

\begin{cases}(x^2 - 6x + 13)y = 20 \\ (y^2 - 6y + 13)z = 20 \\ (z^2 - 6z + 13)x = 20 \end{cases}

6

1

(PAB)=(PBC)=(PCD)=(PDA), P interior of convex ABCD, diagonal bisects area

Suppose that there is a point

P

inside a convex quadrilateral

ABCD

such that the triangles

PAB

,

PBC

,

PCD

,

PDA

have equal areas. Prove that one of the diagonals bisects the area of

ABCD

.

1991 Austrian-Polish Competition

Subcontests

max of f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}

f: A \to A, f(k)\ne g(k) and f(f(f(k))= g(k)$ for k \in A

xy=- 1 (mod z), yz = 1 (mod x), zx = 1 (mod y).

x^2+y^2+z^2 + xy+yz + zx> =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z>0

P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)

(m \choose 2) = 3 (n \choose 4) combinations

P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and midpoints A_1, A_2, A_3, A_1,..

(x^2 - 6x + 13)y = 20,(y^2 - 6y + 13)z = 20, (z^2 - 6z + 13)x = 20

(PAB)=(PBC)=(PCD)=(PDA), P interior of convex ABCD, diagonal bisects area

1991 Austrian-Polish Competition

Subcontests

max of f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}

f: A \to A, f(k)\ne g(k) and f(f(f(k))= g(k)$ for k \in A

xy=- 1 (mod z), yz = 1 (mod x), zx = 1 (mod y).

x^2+y^2+z^2 + xy+yz + zx&gt; =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z&gt;0

P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)

(m \choose 2) = 3 (n \choose 4) combinations

P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and midpoints A_1, A_2, A_3, A_1,..

(x^2 - 6x + 13)y = 20,(y^2 - 6y + 13)z = 20, (z^2 - 6z + 13)x = 20

(PAB)=(PBC)=(PCD)=(PDA), P interior of convex ABCD, diagonal bisects area

x^2+y^2+z^2 + xy+yz + zx> =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z>0