▪ Baltic Way 2011

1.The real numbers satisfy

where is a permutation of . Prove that .
2. Let be a function such that for all integers and , the following holds:

Show that is bounded. 

 

3. A sequence of non-negative integers is such that is the last digit of for all . Is it always true that for some the sequence is periodic? .
4. Let be non-negative reals such that . Prove the inequality

5. Let be a function such that

for all real numbers . Determine . 6. Let be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates , is at least .
7. Let denote the -element set . Let be a subset of in which all six digits appear and in which no three elements together use all these six digits. Determine the largest possible size of .
8. In Greifswald there are three schools called and , each of which is attended by at least one student. Among any three students, one from , one from and one from , there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:
Some student from knows all students from .
Some student from knows all students from .
Some student from knows all students from .
9. Given a rectangular grid, split into squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
All squares touching the border of the grid are coloured black.
No four squares forming a square are coloured in the same colour.
No four squares forming a square are coloured in such a way that only diagonally touching
squares have the same colour.Which grid sizes (with ) have a valid colouring?
10. Two persons play the following game with integers. The initial number is . The players move in turns. Each move consists of subtraction of an integer between and inclusive, or division by , rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?
11. Let and be two diameters of the circle . For an arbitrary point on , let and be the feet of the perpendiculars from to and, respectively. Show that the length of is independent of the choice of .
12. Let be a point inside a square such that is . Determine the angle .
13. Let be an interior point of the convex quadrilateral . Construct triangles and on the outside of the quadrilateral such that the similarities and hold. Let and be the projections of on the lines and , respectively. Prove that if the quadrilateral is cyclic, then

14. The incircle of a triangle touches the sides at , respectively. Let be a point on the incircle such that is a diameter. The lines and intersect at . Prove that .
15. Let be a convex quadrilateral such that . Suppose that a point on the side satisfies the equality

Show that .
16. Let be any integer. Define the sequence by , , and for all

Determine the largest integer for which there exists a prime such that divides .
17. Determine all positive integers such that whenever divides a positive integer , will also divide any integer obtained by rearranging the digits of .
18. Determine all pairs of primes for which both and are perfect squares.
19. Let be a prime number. Show that there is a non-constant arithmetic sequence of positive integers such that the product of the terms of the sequence is a cube.
20. An integer is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers such that there are exactly two balanced numbers among and .