MathDB

1

rearranging reals in a m x n array

Pairwise distinct real numbers are arranged into an

m \times n

rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.

9

1

f(f (x)) = f (x)^2, f (1) = 1, continuous monotonic

Find all continuous monotonic functions

f : R \to R

that satisfy

f (1) = 1

and

f(f (x)) = f (x)^2

for all

x \in R

.

7

1

0<k< n^2/4, k has no prime divisor > n, then k divides n!

Let

k

and

n

be integers with

0 < k < n^2/4

such that k has no prime divisor greater than

n

. Prove that

k

divides

n!

.

5

1

x^2+y^2+u^2+v^2=4, xu+yv+xv +yu=0, xyu+yuv+uvx+vxy=-2, xyuv = -1

Find all real solutions of the system of equations

\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}

3

1

exists a unit square having exactly 0,1 or 4 blue vertices

Each point in space is colored either blue or red. Show that there exists a unit square having exactly

0, 1

or

4

blue vertices.

2

1

a_1P(1) > 2n^2a_o, monic polynomial

The monic polynomial

P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0

of degree

n > 1

has

n

distinct negative roots. Prove that

a_1P(1) > 2n^2a_o

1

3 tangents circles in pairs, similar triangles, angles wanted

A non-right triangle

A_1A_2A_3

is given. Circles

C_1

and

C_2

are tangent at

A_3, C_2

and

C_3

are tangent at

A_1

, and

C_3

and

C_1

are tangent at

A_2

. Points

O_1,O_2,O_3

are the centers of

C_1, C_2, C_3

, respectively. Supposing that the triangles

A_1A_2A_3

and

O_1O_2O_3

are similar, determine their angles.

6

1

R/r of set of tetrahedra with concentric inscribed + circumscribed spheres

Let

M

be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by

r

and

R

respectively, find the possible values of

R/r

over all tetrahedra from

M

.

4

1

m,n and N

Find all triples (m,n,N) of positive integers numbers m,n and N such that

m^N-n^N=2^{100}

with N>1

1986 Austrian-Polish Competition

Subcontests

rearranging reals in a m x n array

f(f (x)) = f (x)^2, f (1) = 1, continuous monotonic

0<k< n^2/4, k has no prime divisor > n, then k divides n!

x^2+y^2+u^2+v^2=4, xu+yv+xv +yu=0, xyu+yuv+uvx+vxy=-2, xyuv = -1

exists a unit square having exactly 0,1 or 4 blue vertices

a_1P(1) > 2n^2a_o, monic polynomial

3 tangents circles in pairs, similar triangles, angles wanted

R/r of set of tetrahedra with concentric inscribed + circumscribed spheres

m,n and N

1986 Austrian-Polish Competition

Subcontests

rearranging reals in a m x n array

f(f (x)) = f (x)^2, f (1) = 1, continuous monotonic

0&lt;k&lt; n^2/4, k has no prime divisor &gt; n, then k divides n!

x^2+y^2+u^2+v^2=4, xu+yv+xv +yu=0, xyu+yuv+uvx+vxy=-2, xyuv = -1

exists a unit square having exactly 0,1 or 4 blue vertices

a_1P(1) &gt; 2n^2a_o, monic polynomial

3 tangents circles in pairs, similar triangles, angles wanted

R/r of set of tetrahedra with concentric inscribed + circumscribed spheres

m,n and N

0<k< n^2/4, k has no prime divisor > n, then k divides n!

a_1P(1) > 2n^2a_o, monic polynomial