Tournament of Towns in Toronto

Practice session October 03, 2015.
Discrete continuity. Continuity. Method of small perturbations.
  1. Two dogs want to divide a string of 30 sausages, 10 made of pork and 20 made of beef. Every dog wants to have half of each kind of sausages. What is the minimal number of cuts they should make?
  2. Several kindergarten children stand in a line, facing a teacher. On the teacher's command to turn left some of them indeed turned left, but some turned right and some even turned around. Is it always possible to place a new girl into the line so that on her both sides the number of children facing her is the same?
  3. Does there exist 100 consecutive numbers among which there are exactly 10 primes?
  4. A white or a black figure is placed on each square of a chess board. Is it always possible to split the board into 2 connected parts by lines along the boundaries of squares so that each part contains exactly half of black and half of white figures? (The number of black figures could be any even number from 0 to 64).
  5. At 6 am in the morning a monk started his way down from the monastery to the village market. At noon he reached the market, bought supplies and stayed for a night in the village. Next day at 6 am he started to the monastery. The way back with heavy load was much harder and he stopped often to rest. He reached the place only at sunset which was at 8 pm. Prove that there was a time when the monk was at the same point on his way down and up. There is only one path between the the monastery and the village.
  6. Around a circular fountain a dog is chasing a cat with a sausage. At some moment the sausage is dropped but chasing continues. Prove that there was a moment when the distances from the cat and the dog to the sausage were the same.
  7. At two opposite corners of $10\text{m}\times 10\text{m}$ pond there are two geese. After some time they were noticed in two other opposite corners. Prove that there was a moment when the distance between their beaks was exactly 13\,m.
  8. On a board a teacher wrote an equation $x^2 -3x+8=0$. One by one students went to the board and each changed by 1 either a coefficient at $x$ or a free term. In the result on the board appeared equation $x^2+7x- 1=0$. Prove that at some moment on the board was equation with integer roots.
  9. Solve inequality $ax^2 +2x+c<0$, if $ac>1$, $c<4(1-a)$.
  10. Does there exist a non equilateral triangle such that the smallest median equals the largest altitude?
  11. Is it true that any convex polygon can be split by a straight line into two polygons with equal area and perimeter?
  12. Does there exist a tetrahedron such that each face is an obtuse triangle?
  13. A number of points are marked on a plane. Prove that one can draw a straight line so that distances from all points to this line are distinct.
  14. Meeting of eleven gangsters ended with a firefight: each gangster shot the most distant person and all shots were fired simultaneously. Given that the distances between any two gangsters are distinct how many gangsters could survive? List all the possibilities. Each gangster shoots once and kills if aimed.
  15. (Tournament of Towns 1997, Spring , A-Level, Junior) On a plate there are 25 pieces of cheese of distinct mass. Is it always possible to cut one piece in two and divide 26 pieces between two plates so that the number of pieces and the total mass of cheese on each plate would be the same?
  16. (Tournament of Towns, Fall 2002, O-Level, Senior) Each subsequent number of infinite sequence of positive integers equals the previous term plus one of it not zero digits. Prove that the sequence contains an even number.
  17. (Tournament of Towns, Spring 2011, A-Level, Senior) Does there exist a convex $n$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ if
    1. $n= 2011$;
    2. $n = 2012$?
  18. (Tournament of Towns, Spring 2004, O-Level, Senior) Perimeter of a convex quadrilateral is 2004 and one of its diagonals is 1001. Can another diagonal be 1? 2? 1001?
  19. (Tournament of Towns, Spring 2013, A-Level, Junior) Two teams A and B play a school ping pong tournament. The team A consists of $m$ students, and the team B consists of $n$ students where $m\ne n$. There is only one ping pong table to play and the tournament is organized as follows. Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

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Tournament of the Towns is held in Toronto since 1996.
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