MathDB

1

p(x^3+1)=p(x+1)^3

p(x)

satisfying

p(x^3+1)=p(x+1)^3

for all

x

.

Problem 5

1

two-colored 3D space, tetrahedron

The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.

Problem 4

1

point moves along ray, line goes through fixed point

Let

A,B,C

be non-collinear points. For each point

D

of the ray

AC

, we denote by

E

and

F

the points of tangency of the incircle of

\triangle ABD

with

AB

and

AD

, respectively. Prove that, as point

D

moves along the ray

AC

, the line

EF

passes through a fixed point.

Problem 2

1

2^(2^n)+1 and 2^(p-1)+1

Let

n

and

k

be natural numbers and

p

a prime number. Prove that if

k

is the exact exponent of

p

in

2^{2^n}+1

(i.e.

p^k

divides

2^{2^n}+1

, but

p^{k+1}

does not), then

k

is also the exact exponent of

p

in

2^{p-1}-1

.

Problem 1

1

parameter, quartic has four roots

Find all real parameters

q

for which there is a

p\in[0,1]

such that the equation

x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0

has four real roots.

Problem 3

1

tetrahedron inequality, face areas

Let

M

be an arbitrary interior point of a tetrahedron

ABCD

, and let

S_A,S_B,S_C,S_D

be the areas of the faces

BCD,ACD,ABD,ABC

, respectively. Prove that

S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,

where

V

is the volume of

ABCD

. When does equality hold?

1988 Bulgaria National Olympiad

Subcontests

p(x^3+1)=p(x+1)^3

two-colored 3D space, tetrahedron

point moves along ray, line goes through fixed point

2^(2^n)+1 and 2^(p-1)+1

parameter, quartic has four roots

tetrahedron inequality, face areas