IMC training 2016 (3)

Methods of Counting (2)

Problem 1. Find the number of pairs (x;y) of integers such that |x|+|y|\le 1000.

Problem 2. How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?

Problem 3. Let x=.1234567891011…998999, where the digits are obtained by writing the integers 1 through 999 in order. Find the {{1983}^{rd}} digit to the right of the decimal point.

Problem 4. A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

Problem 5.  Find the number of sets {a,b,c} of three distinct positive integers with the property that the product of a,b, and c is equal to the product of 11,21,31,41,51, and 61.

Problem 6. Find the number of five-digit positive integers, n, that satisfy the following conditions:

(a) the number n is divisible by 5,

(b) the first and last digits of n are equal, and

(c) the sum of the digits of n is divisible by 5.

Problem 7. Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.

Problem 8. A square 8\times 8 is divided into 64 unit squares in the usual manner. Each of the 81 vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

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