MathDB

1

rectangular blocks of size a x b x c, to fill up a cubic box of edge 10

(a, b, c)

of positive integers, not all equal, We are given sufficiently many rectangular blocks of size

a \times b \times c

. We use these blocks to fill up a cubic box of edge

10

. (a) Assume we have used at least

100

blocks. Show that there are two blocks, one of which is a translate of the other. (b) Find a number smaller than

100

(the smaller, the better) for which the above statement still holds.

8

1

P(x^2 + 1) = P(x)^2 - 1, polynomial of degree 998

Show that there is no polynomial

P(x)

of degree

998

with real coefficients which satisfies

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

for all

x

.

4

1

- 1<=xy + yz + zt + tx <=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1

Real numbers

x,y,z, t

satisfy

x + y + z +t = 0

and

x^2+ y^2+ z^2+t^2 = 1

.
Prove that

- 1 \le xy + yz + zt + tx \le 0

.

6

1

x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 system

Given natural numbers

n > k > 1

, find all real solutions

x_1,..., x_n

of the system

x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2

for 1

\le i \le n

. Here

x_{n+i} = x_i

for all

i

.

1

exist exactly 3^{k-1} natural numbers n with the 4 following properties

Let

k \ge 1

be a positive integer. Prove that there exist exactly

3^{k-1}

natural numbers

n

with the following properties: (i)

n

has exactly

k

digits (in decimal representation), (ii) all the digits of

n

are odd, (iii)

n

is divisible by

5

, (iv) the number

m = n/5

has

k

odd digits

5

1

sphere S divides every edge of a convex polyhedron P into 3 equal parts.

A sphere

S

divides every edge of a convex polyhedron

P

into three equal parts. Show that there exists a sphere tangent to all the edges of

P

.

3

1

Polynomial and equation

The polynomials

P_{n}(x)

are defined by

P_{0}(x)=0,P_{1}(x)=x

and P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) n\geq 2 For every natural number

n\geq 1

, find all real numbers

x

satisfying the equation

P_{n}(x)=0

.

7

1

k!+48=48(k+1)^m

Prove there are no such integers

k, m

which satisfy

k \ge 0, m \ge 0

and

k!+48=48(k+1)^m

.

2

1

A convex haxagon and three conditions.

A convex hexagon

ABCDEF

satisfies the following conditions: 1)

AB\parallel DE

,

BC\parallel EF

, and

CD\parallel FA

. 2) The distances between these pairs of parallel lines are the same. 3) \angle FAB \equal{} \angle CDE \equal{} 90^\circ Prove that the diagonals

BE

and

CF

of the hexagon intersect with angle

45

degrees.

\bullet

Thank you dear Babis Stergiou for your translation. :P

1996 Austrian-Polish Competition

Subcontests

rectangular blocks of size a x b x c, to fill up a cubic box of edge 10

P(x^2 + 1) = P(x)^2 - 1, polynomial of degree 998

- 1<=xy + yz + zt + tx <=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1

x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 system

exist exactly 3^{k-1} natural numbers n with the 4 following properties

sphere S divides every edge of a convex polyhedron P into 3 equal parts.

Polynomial and equation

k!+48=48(k+1)^m

A convex haxagon and three conditions.

1996 Austrian-Polish Competition

Subcontests

rectangular blocks of size a x b x c, to fill up a cubic box of edge 10

P(x^2 + 1) = P(x)^2 - 1, polynomial of degree 998

- 1&lt;=xy + yz + zt + tx &lt;=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1

x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 system

exist exactly 3^{k-1} natural numbers n with the 4 following properties

sphere S divides every edge of a convex polyhedron P into 3 equal parts.

Polynomial and equation

k!+48=48(k+1)^m

A convex haxagon and three conditions.

- 1<=xy + yz + zt + tx <=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1