MathDB

1

g(x) = f(x) h(x), polynomials

f(x)

and

g(x)

be polynomials with integer coefficients. Prove that if for every integer value

n

the number

g(n)

is divisible by the number

f(n)

, then

g(x) = f(x)\cdot h(x)

, where

h(x)

is a polynomial,. Show with an example that the coefficients of the polynomial

h(x)

do not have to be integer.

5

1

equal sums of colored areas, spherw inscribed in convex polyhedron

Prove that if a sphere can be inscribed in a convex polyhedron and each face of this polyhedron can be painted in one of two colors such that any two faces sharing a common edge are of different colors, then the sum of the areas of the faces of one color is equal to the sum of the areas of the faces of the other color.

4

1

( sum a_i x_i^2 )^2 <= sum a_i x_i^4

Prove that the non-negative numbers

a_1, a_2, \ldots, a_n

(

n = 1, 2, \ldots

) satisfy the inequality

x_1, x_2, \ldots, x_n

for any real numbers

\left( \sum_{i=1}^n a_i x_i^2 \right)^2 \leq \sum_{i=1}^n a_i x_i^4.

it is necessary and sufficient that

\sum_{i=1}^n a_i \leq 1

.

3

1

probability to wash the dishes

In a certain family, a husband and wife made the following agreement: If the wife washes the dishes one day, the husband washes the dishes the next day. However, if the husband washes the dishes one day, then who washes the dishes the next day is decided by drawing a coin. Let

p_n

denote the probability of the event that the husband washes the dishes on the

n

-th day of the contract. Prove that there is a limit

\lim_{n\to \infty} p_n

and calculate it. We assume

p_1 = \frac{1}{2}

.

2

1

if 2 quads are tangential, so is a third one

In the convex quadrilateral

ABCD

, the corresponding points

M

and

N

are chosen on the adjacent sides

\overline{AB}

and

\overline{BC}

and the intersection point of the segments

AN

and

GM

is marked by 0. Prove that if circles can be inscribed in the quadrilaterals

AOCD

and

BMON

, then a circle can also be inscribed in the quadrilateral

ABCD

.

1

W(x) = x^4 + ax^3 + bx + cx + d

The polynomial

W(x) = x^4 + ax^3 + bx + cx + d

is given. Prove that if the equation

W(x) = 0

has four real roots, then for there to exist

m

such that

W(x+m) = x^4+px^2+q

, it is necessary and it is enough that the sum of certain two roots of the equation

W(x) = 0

equals the sum of the remaining ones.

1975 Poland - Second Round

Subcontests

g(x) = f(x) h(x), polynomials

equal sums of colored areas, spherw inscribed in convex polyhedron

( sum a_i x_i^2 )^2 <= sum a_i x_i^4

probability to wash the dishes

if 2 quads are tangential, so is a third one

W(x) = x^4 + ax^3 + bx + cx + d

1975 Poland - Second Round

Subcontests

g(x) = f(x) h(x), polynomials

equal sums of colored areas, spherw inscribed in convex polyhedron

( sum a_i x_i^2 )^2 &lt;= sum a_i x_i^4

probability to wash the dishes

if 2 quads are tangential, so is a third one

W(x) = x^4 + ax^3 + bx + cx + d

( sum a_i x_i^2 )^2 <= sum a_i x_i^4