The 20-th Austrian–Polish Mathematics Competition Austria, June 25–27, 1997

1. Lines ℓ_{1} and ℓ_{2} intersect in point P. Circles S_{1} and S_{2} are externally tangent in P; the line ℓ_{1} is their common tangent in this point. Similarly, circles T_{1} and T_{2} are externally tangent in point P and ℓ_{2} is their common tangent in this point. Circles S_{1} and T_{1} intersect in points P and A. Similarly, circles S_{1} and T_{2} intersect in points P and B, circles S_{2} and T_{2} intersect in points P and C and circles S_{2} and T_{1} intersect in points P and D. Prove that the points A, B, C and D are concyclic if and only if the lines ℓ_{1} and ℓ_{2} are perpendicular.

2. Given is an n × m chessboard. Each field is assign a pair of coordinates (x, y) where 1 ≤ x ≤ m and 1 ≤ y ≤ n. Let p and q be positive integers. A pawn standing on the field (x, y) may be moved to the field (x′, y′) if and only if

|x − x′| = p and |y − y′| = q .

There is a pawn on each field. We want to move each pawn at the same time such that after the move there is still only one pawn on each field. In how many ways can this be done?

3. There are 97 numbers written on the blackboard: 48, 24, 16, …, 48/97, that is, rational numbers 48/k for k = 1, 2, …, 97. In each step we choose two numbers a and b from the blackboard, remove them and write 2ab − a − b + 1 instead. After 96 steps there will remain exactly one number on the blackboard. Determine the set of these numbers.

4. In a convex quadrilateral ABCD the sides AB and CD are parallel. The diagonals AC and BD intersect in point E. F is the point of intersection of altitudes of triangle EBC, and G is the point of intersection of altitudes of triangle EAD. Prove that the midpoint of GF lies on the line passing through E and perpendicular to AB.

5. Let p_{1}, p_{2}, p_{3}, p_{4} be four distinct primes. Prove that there is no polynomial Q(x) = ax^{3} + bx^{2} + cx + d

with integer coefficients such that |Q(p_{1})| = |Q(p_{2})| = |Q(p_{3})| = |Q(p_{4})| = 3 .

6. Prove that there does not exist function f: Z → Z such that f(x + f(y)) = f(x) − y for all integers x, y. Note: Z denotes the set of integers.

7. a) Prove that for any real numbers p, q the following inequality holds:

p^{2} + q^{2} + 1 > p(q + 1) .

b) Determine the biggest real number b such that for all real numbers p, q the following inequality holds: p^{2} + q^{2} + 1 ≥ bp(q + 1) .

c) Determine the biggest real number c such that for all integer numbers p, q the following inequality holds:

p^{2} + q^{2} + 1 ≥ cp(q + 1) .

8. Let n be a positive integer and let M be an n-element set. Determine the biggest positive integer k with the following property: There exists a k-element family K consisting of 3-element subsets of the set M, such that each two sets belonging to K have nonempty intersection.

9. Given is a parallelepiped P. Let V_{P} be its volume, S_{P} the area of its surface and L_{P} the sum of the lengths of its edges. For a real number t ≥ 0 let P_{t} be the solid consisting of all points X whose distance from P is not bigger than t. Prove that the volume of the solid P_{t} is given by the formula

V (P_{t}) = V_{P} + S_{P} t + \frac{π}{4} L_{P} t^{2} + \frac{4π}{3} t^{3} .

Explanation: A distance from point X to parallelepiped P is not bigger than t if there exists point Y belonging to parallelepiped P such that |XY | ≤ t.