MathDB

1

vector w = (a, b, c) such that a^2 + b^2 + c^2<= 48

Consider the cube with the vertices at the points

(\pm 1, \pm 1, \pm 1)

. Let

V_1,...,V_{95}

be arbitrary points within this cube. Denote

v_i = \overrightarrow{OV_i}

, where

O = (0,0,0)

is the origin. Consider the

2^{95}

vectors of the form

s_1v_1 + s_2v_2 +...+ s_{95}v_{95}

, where

s_i = \pm 1

. (a) If

d = 48

, prove that among these vectors there is a vector

w = (a, b, c)

such that

a^2 + b^2 + c^2 \le 48

. (b) Find a smaller

d

(the smaller, the better) with the same property.

1

a_n = a_{n-1} + a_{n-2} , nxn system

Determine all real solutions

(a_1,...,a_n)

of the following system of equations:

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

7

1

3y^4 + 4cy^3 + 2xy + 48 = 0

Consider the equation

3y^4 + 4cy^3 + 2xy + 48 = 0

, where

c

is an integer parameter. Determine all values of

c

for which the number of integral solutions

(x,y)

satisfying the conditions (i) and (ii) is maximal: (i)

|x|

is a square of an integer; (ii)

y

is a squarefree number.

4

1

P(x)^2 + P(1/x)^2= P(x^2)P(1/x^2)

Determine all polynomials

P(x)

with real coefficients such that

P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)

for all

x

.

6

1

E_i \cap E_j \ne\varnothing, 4mountain trips for n Alpine Club's members

The Alpine Club organizes four mountain trips for its

n

members. Let

E_1, E_2, E_3, E_4

be the teams participating in these trips. In how many ways can these teams be formed so as to satisfy

E_1 \cap E_2 \ne\varnothing

,

E_2 \cap E_3 \ne\varnothing

,

E_3 \cap E_4 \ne\varnothing

?

3

1

Q(y) R(y)= P(5y^2) when P(x) = x^4 + x^3 + x^2 + x + 1

Let

P(x) = x^4 + x^3 + x^2 + x + 1

. Show that there exist two non-constant polynomials

Q(y)

and

R(y)

with integer coefficients such that for all

Q(y) \cdot R(y)= P(5y^2)

for all

y

.

2

1

subset Y of X with property that there is no disk K such that K\cap X = Y

Let

X= \{A_1, A_2, A_3, A_4\}

be a set of four distinct points in the plane. Show that there exists a subset

Y

of

X

with the property that there is no (closed) disk

K

such that

K\cap X = Y

.

9

1

(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n) >= ...

Prove that for all positive integers

n,m

and all real numbers

x, y > 0

the following inequality holds:

(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n)\\ \\ \ge nm(x^{n+m-1}y + xy^{n+m-1}).

5

1

equilateral triangle

ABC

is an equilateral triangle.

A_{1}, B_{1}, C_{1}

are the midpoints of

BC, CA, AB

respectively.

p

is an arbitrary line through

A_{1}

.

q

and

r

are lines parallel to

p

through

B_{1}

and

C_{1}

respectively.

p

meets the line

B_{1}C_{1}

at

A_{2}

. Similarly,

q

meets

C_{1}A_{1}

at

B_{2}

, and

r

meets

A_{1}B_{1}

at

C_{2}

. Show that the lines

AA_{2}, BB_{2}, CC_{2}

meet at some point

X

, and that

X

lies on the circumcircle of

ABC

.

1995 Austrian-Polish Competition

Subcontests

vector w = (a, b, c) such that a^2 + b^2 + c^2<= 48

a_n = a_{n-1} + a_{n-2} , nxn system

3y^4 + 4cy^3 + 2xy + 48 = 0

P(x)^2 + P(1/x)^2= P(x^2)P(1/x^2)

E_i \cap E_j \ne\varnothing, 4mountain trips for n Alpine Club's members

Q(y) R(y)= P(5y^2) when P(x) = x^4 + x^3 + x^2 + x + 1

subset Y of X with property that there is no disk K such that K\cap X = Y

(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n) >= ...

equilateral triangle

1995 Austrian-Polish Competition

Subcontests

vector w = (a, b, c) such that a^2 + b^2 + c^2&lt;= 48

a_n = a_{n-1} + a_{n-2} , nxn system

3y^4 + 4cy^3 + 2xy + 48 = 0

P(x)^2 + P(1/x)^2= P(x^2)P(1/x^2)

E_i \cap E_j \ne\varnothing, 4mountain trips for n Alpine Club's members

Q(y) R(y)= P(5y^2) when P(x) = x^4 + x^3 + x^2 + x + 1

subset Y of X with property that there is no disk K such that K\cap X = Y

(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n) &gt;= ...

equilateral triangle

vector w = (a, b, c) such that a^2 + b^2 + c^2<= 48

(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n) >= ...