The 22nd Austrian–Polish Mathematics Competition Austria, June 30 – July 2, 1999

1. Let n be a positive integer and M = 1, 2, …, n. Find the number of ordered 6-tuples (A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}) which satisfy the following two conditions:

a) sets A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} (not necessarily different) are subsets of M

b) each element of M belongs either to exactly three subsets or to exactly six subsets or does not belong to any subset A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}.

2. Find the largest real number C_{1} and the smallest real number C_{2} such that for all real numbers a, b, c, d, e the following inequalities hold

C_{1} < \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + d} + \frac{d}{d + e} + \frac{e}{e + a} < C_{2} .

3. Let n ≥ 2 be a given integer. Determine all systems of n functions (f_{1}, …, f_{n}) where f_{i} : R → R i = 1, …, n such that for all x, y ∈ R the following equalities hold

f_{1}(x) − f_{2}(x)f_{2}(y) + f_{1}(y) = 0
f_{2}(x^{2}) − f_{3}(x)f_{3}(y) + f_{2}(y^{2}) = 0
. . . . . .
f_{k}(x^{k}) − f_{k+1}(x)f_{k+1}(y) + f_{k}(y^{k}) = 0
. . . . . .
f_{n}(x^{n}) − f_{1}(x)f_{1}(y) + f_{n}(y^{n}) = 0 .

4. Through a point P, which lies inside the triangle ABC, are drawn three straight lines k, l, m in such a way that:

a) k meets the lines AB and AC in A_{1} and in A_{2} (A_{1} ≠ A_{2}) respectively and P A_{1} = P A_{2} ,

b) similarly l meets the lines BC and BA in B_{1} and in B_{2} (A_{1} ≠ A_{2}) respectively and P B_{1} = P B_{2} ,

c) and similarly m meets the lines CA and CB in C_{1} and in C_{2} (C_{1} ≠ C_{2}) respectively and P C_{1} = P C_{2} . Prove that the lines k, l, m are uniquely determined by the conditions a),b),c). Find the point P (and prove that there exists exactly one such point) for which the triangles AA_{1}A_{2}, BB_{1}B_{2}, and CC_{1}C_{2} have the same area.

5. A sequence of integers (a_{n}) satisfies the following recursive equation

a_{n+1} = a_{n}^{3} + 1999 for n = 1, 2, ….

Prove that there exists at most one such n for which a_{n} is the square of an integer.

6. Solve the following system of equations

x_{n}^{2} + x_{n}x_{n−1} + x_{n−1}^{4} = 1 for n = 1, 2, …, 1999
x_{0} = x_{1999}

in the set of nonnegative real numbers.

7. Find all pairs (x, y) of positive integers such that

x^{x+y} = y^{y−x} .

8. Let g be a given straight line and let the points P, Q, R all lie on the same side of the line g. The points M, N lie on the line g and satisfy P M ⊥ g and QN ⊥ g. The point S lies between the lines P M and QN and additionaly satisfies P M = P S and QN = QS. The bisectors of SM and SN meet in the point R. The line RS intersects the circumcircle of the triangle P QR in T ≠ R. Prove that S is the midpoint of the segment RT.

9. A point in the plane with both integer cartesian coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold:

a) the endpoints of each selected segment are lattice points,

b) each selected segment is parallel to a coordinate axis, or to the line y = x, or to the line y = −x,

c) each selected segment contains exactly five lattice points and all of them are selected,

d) each two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and selected segment is a position. Prove or disprove that there exists an initial position such that the game has infinitely many moves.