The 18-th Austrian–Polish Mathematics Competition Austria, June 28–30, 1995

1. Let n be a natural number. Determine all solutions (a_{1}, …, a_{n}) of the following system of equations

\{ \begin{matrix}a_{3} = a_{2} + a_{1} \\ a_{4} = a_{3} + a_{2} \\ ⋯ \\ a_{n} = a_{n−1} + a_{n−2} \\ a_{1} = a_{n} + a_{n−1} \\ a_{2} = a_{1} + a_{n}, \end{matrix}

where a_{1}, …, a_{n} are real numbers.

2. Let A_{1}, A_{2}, A_{3}, A_{4} be four distinct points on the plane and let X = {A_{1}, A_{2}, A_{3}, A_{4}}. Prove that there exists a subset Y of X with the following property:

There does not exist a closed disc K, such that K ∩ X = Y.

Note: All the points on the circle belong to the closed disc.

3. Let P (x) = x^{4} + x^{3} + x^{2} + x + 1. Prove that there exist two polynomials Q(y) and R(y) with degree greater than or equal to 1, with integer coefficients, such that for all y

Q(y) · R(y) = P (5y^{2}).

4. Determine all polynomials P (x) with real coefficients, such that for all x ≠ 0

(P (x))^{2} + (P (1/x))^{2} = P (x^{2}) P (1/x^{2}).

5. Given is an equilateral triangle ABC. Let A_{1} be the midpoint of BC, B_{1} the midpoint of AC and C_{1} the midpoint of AB. Let p, q, r be three distinct parallel lines, such that the point A_{1} lies on the line p, the point B_{1} lies on the line q and the point C_{1} lies on the line r. The line p intersects the line B_{1}C_{1} in point A_{2}, the line q intersects the line A_{1}C_{1} in point B_{2} and the line r intersects the line A_{1}B_{1} in point C_{2}. Prove that the lines AA_{2}, BB_{2}, CC_{2} intersect in one point D and the point D lies on the circumcircle of the triangle ABC.

6. The Alpine Club, consisting of n members, organizes four mountain trips for their members. Let E_{1}, E_{2}, E_{3}, E_{4} be the teams participating in these trips. In how many ways one can choose the teams with the condition that

E_{1} ∩ E_{2} ≠ ∅,  E_{2} ∩ E_{3} ≠ ∅,  E_{3} ∩ E_{4} ≠ ∅?

7. For every integer c we consider the equation

3y^{4} + 4cy^{3} + 2xy + 48 = 0.

In this equation the unknowns x and y are integers. Determine all integers c, such that the number of the integral solutions (x, y), satisfying the additional conditions (A) and (B), is maximal. (A) The number |x| is a square of an integer. (B) The number y is squarefree, i.e. there doesn’t exist a prime p, such that p^{2} is a divisor of y.

8. Consider the cube whose eight vertices have the coefficients (±1, ±1, ±1), i.e. the set of the points

{(x, y, z): |x| ≤ 1, |y| ≤ 1, |z| ≤ 1}.

Let V_{1}, …, V_{95} be any points of this cube. Denote by v_{i} the vector from the point (0, 0, 0) to the point V_{i}. Consider the 2^{95} vectors of the form s_{1}v_{1} + s_{2}v_{2} + ⋯ + s_{95}v_{95}, where s_{i} = +1 or s_{i} = −1.

a) Let d = 48. Prove that among these vectors there exists a vector w = (a, b, c), such that a^{2} + b^{2} + c^{2} ≤ d.

b) Find a number d < 48 with the same property. Note: The less the number d is obtained, the higher the solution will be scored.

9. Prove that for all natural numbers n, m ≥ 1 and for all positive real numbers x, y the following inequality holds:

(n − 1)(m − 1)(x^{n+m} + y^{n+m}) + (n + m − 1)(x^{n}y^{m} + x^{m}y^{n}) ≥ nm(x^{n+m−1}y + xy^{n+m−1}).