44-th Mathematical Olympiad in Poland First Round, September–December, 1992

1. Solve the following equation in real numbers:

\frac{(x^{2} − 1)(|x| + 1)}{x + sgn x} = [x + 1].

2. Given is a natural number n ≥ 3. Solve the system of equations:

tan x_{1} + 3 cot x_{1} = 2 tan x_{2}
tan x_{2} + 3 cot x_{2} = 2 tan x_{3}
. . . . . . . . . . . . . . . . . . . . . . . . . . .
tan x_{n} + 3 cot x_{n} = 2 tan x_{1} .

3. Given is a hexagon ABCDEF with a center of symmetry. The lines AB and EF meet at the point A′, the lines BC and AF meet at the point B′, and the lines AB and CD meet at the point C′. Prove that AB · BC · CD = AA′ · BB′ · CC′.

4. Determine all functions f: R ⟶ R such that

f(x + y) − f(x − y) = f(x) · f(y) for x, y ∈ R.

5. Given is a halfplane with points A and C on its edge. For every point B on this halfplane consider the squares ABKL and BCMN lying outside of the triangle ABC. Prove that all the lines LM (as the point B varies) have a common point.

6. The sequence (x_{n}) is determined by the conditions:

x_{0} = 1992, x_{n} = −\frac{1992}{n} · ∑_{k=0}^{n−1} x_{k} for n ≥ 1.

Find ∑_{n=0}^{1992} 2^{n}x_{n} .

7. Given are the points A_{0} = (0, 0, 0), A_{1} = (1, 0, 0), A_{2} = (0, 1, 0), A_{3} = (0, 0, 1) in the space. Let P_{ij} (i, j ∈

\{0, 1, 2, 3\}) be the point determined by the equality: \overrightarrow{A_{0}P_{ij}} = \overrightarrow{A_{i}A_{j}}. Find the volume of the smallest convex polyhedron which contains all the points P_{ij}.

8. Given is a positive integer n ≥ 2. Determine the maximum value of the sum of natural numbers k_{1}, k_{2}, …, k_{n}

satisfying the condition:

k_{1}^{3} + k_{2}^{3} + ⋯ + k_{n}^{3} ≤ 7n .

9. Prove that for all real numbers a, b, c the inequality

(a^{2} + b^{2} − c^{2})(b^{2} + c^{2} − a^{2})(c^{2} + a^{2} − b^{2}) ≤ (a + b − c)^{2}(b + c − a)^{2}(c + a − b)^{2}

holds.

10. Let C be a cube and let f: C → C be a surjection with

|P Q| ≥ |f(P)f(Q)|

for all P, Q ∈ C. Prove that f is an isometry.

11. Given is an n × n chessboard. With the same probability, we put six pawns on its six cells. Let p_{n} denotes the probability that there exists a row or a column containing at least two pawns. Find lim_{n→∞} np_{n}.

12. Prove that the polynomial x^{n} + 4 can be expressed a product of two polynomials (each with degree less than n) with integer coefficients, if and only if n is divisible by 4.