Tournament of Towns in Toronto

Practice session January 25, 2014. Combinatorics
  1. At a meeting, each of the 25 journalists shook hands with exactly $n$ of his or her colleagues. Can it happen that
    1. $n=5$?
    2. $n=4$?
  2. There are 28 students in Miss Brown's class. Today, every girl shook hands with 4 boys and every boy shook hands with 3 girls. How many boys and how many girls are there in Miss Brown's class?
  3. Consider the numbers from 1 to 1000 inclusive. How many of them contain the digit 3? How many contain both the digit 1 and the digit 2?
  4. For every two-digit number, consider the product of its digits. What is the the total sum of all these products? Also answer the question for 3 digit numbers.
  5. City $A$ is connected with city $B$ by 4 roads, city $B$ is connected with city $C$ by 2 roads, and city $C$ is connected with city $A$ by 3 roads. What is the number of ways to make a trip $A\to B\to C \to A$? To make a trip from $A$ to $C$? (Loops are not allowed).
  6. Consider all seven digit numbers. Are there more numbers that contain the digit 1 or the ones that do not?
  7. There are 11 red and 17 blue points on a circle. Every two points are connected by segments. How many segments have ends of the different colours? How many segments have both ends of the same colour?
  8. What is the number of diagonals of a convex $n$-gon?
  9. What is the number of ways to place 8 rooks on a chessboard, so that no two rooks attack one another?
  10. A fence has 20 boards. Each board should be painted in one of three colours, either blue, green or yellow, so that no two adjacent boards have the same colour.
    1. In how many ways can this be done?
    2. The same question given some board must be blue.
  11. A kitchen has five independent lights. In how many ways it can be illuminated?
  12. Four friends came to an ice-cream stand. In how many ways can they stand in line?
  13. After the semifinals, only seven candidates were in the running to win medals. In how many ways can the Gold, Silver and Bronze medals be distributed?
  14. In a ski training camp, there were 12 athletes. How many ways are there to select a team of 4 athletes for the winter olympics?
  15. In how many ways, can one arrange 9 different textbooks on a shelf so that the Chemistry and Calculus textbooks are placed next to each other?
  16. There are 2 girls and 7 boys in a chess club. A team of 5 participants should be chosen for a tournament, consisting of at least one girl. In how many ways can this be done?
  17. Only 12 of 17 candidates who passed the final test of the NASA program will go to a space journey. What is the number of ways to select a group given that two candidates psychologically incompatible?
  18. Peter and Basil have 5 and 7 comics books respectively. In how many ways can they trade two books for two books with each other? All books are different.
  19. The menu at Mike's school cafeteria consists of 10 different options (items) and never changes. Mike wants to have a different meal each day (consisting of one or several options) for as many days as possible.
    1. What is the maximum number of days Mike can do it?
    2. What is the total number of items Make eats during this time?
  20. How many "words" can be created by permutation of the following letters:
    1. TRIANGLE
    2. POLYGON
    3. QUADRILATERAL?
  21. There are two parallel lines. Eight points are marked on one line, on six on the other. What is the number of a) triangles b) quadrilaterals whose vertices belong to the marked points?
  22. A necklace consists of 5 beads. Each bead is either green or blue. What is the number of different necklaces?
  23. Find the coefficients at $x^{17}$ and $x^{18}$ in the expansion of the expression $(1+x^5+x^7)^{20}$.
  24. In how many ways can one place 48 eggs into 6 baskets?
    1. If baskets are allowed to be empty,
    2. If no basket is allowed to be empty.
Problems for whole family
  1. A magician shows a trick. He is blindfolded and wears gloves. There are two tables and 20 identical coins. He asks someone from audience to arrange the coins on one of the tables so that 10 coins are head up and and 10 coins are head down. The magician can flip any number of coins and move any number of coins to the other table. The trick magician wants to show is to get on both tables the same number of head up coins. How can he do it? (The magician can not distinguish the sides of coin by touch) .
  2. According to a prescription, every morning a patient must take two pills, one from the white bottle and one from the black bottle. The amount of pills in each bottle is the same. Today, the patient dropped one pill from the white bottle and accidentally two pills from the black bottle (instead of one). All three pills look identical. How should he take the medicine in order to follow the prescription? (The patient must use all amount of pills).

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