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. Continue reading “IMC training 2016 (3)”

IMC training 2016 (2)

Methods of Counting

Problem 1. How many subsets are there in a set of size 10?

Problem 2. Find the number of squares contained in an 10\times 10 squares array.

Problem 3. A team is to be chosen from 4 girls and 6 boys. The only requirement is that it must contain at least 2 girls. Find the number of different teams that may be chosen.

Problem 4. You wish to give your Aunt Mollie a basket of fruit. In your refrigerator you have six oranges and nine apples. The only requirement is that there must be at least one piece of fruit in the basket (that is, an empty basket of fruit is not allowed). How many different baskets of fruit are possible?

Problem 5. Find the number of ways 30 identical pencils can be distributed among three girls so that each gets at least 1 pencil.

Problem 6. There 7 boys and 3 girls in a gathering. In how many ways can they be arranged in a row so that

1) the 3 girls form a single block?

2) the two end-possitions are occupied by boys and no girls are adjacent?

Problem 7. Between 20000 and 70000, find the number of even integers in which no digit is repeated. Continue reading “IMC training 2016 (2)”

IMC training 2016 (1)

The arithmetic of integers

Problem 1. The positive integers from 1 to 12 have been divided into six pairs so that the sum of the two numbers in each pair is a distinct prime number. Find the largest of these prime number.

Problem 2. The sum of the squares of three prime numbers is 5070. Find the product of these three prime numbers.

Problem 3. A number is said to be strange if in its prime factorization, the power of each prime number is odd. For istance, 22,23 and 24 form a block of three consecutive strange numbers because 22={{2}^{1}}\times {{11}^{1}},23={{23}^{1}},24={{2}^{3}}\times {{3}^{1}}. Find the greatest length of a block of consecutive strange numbers.

Problem 4. Find the number of consecutive 0s at the end of 2003!=1\times 2\times 3\times ...\times 2003.

Problem 5. Find the smallest positive integer which is 2 times the square of some positive integer and also 5 times the fifth power of some other positive integer.

Problem 6. Sum of seven consecutive positive integer is the cube of an integer and the sum of the middle three numbers is the square of an integer. Find the smallest possible value of the middle number.

Problem 7. Some factors in the product 1\times 2\times 3\times 4\times ...\times 27 are to be removed so that the product of the remaining factors is the square of an integer. Find the minimum number of factors that must be removed. Continue reading “IMC training 2016 (1)”

On the elementary symmetric functions of 1, 1⁄2, …, 1⁄n

Cho số nguyên dương nS(k,n) là hàm đối xứng sơ cấp thứ k của \displaystyle 1,\frac{1}{2},\ldots,\frac{1}{n}, nghĩa là \displaystyle S(k,n)=\sum_{1\leq i_1<i_2<\cdots<i_k\leq n}\frac{1}{i_1i_2\ldots i_k}.

Ta biết rằng nếu n>1 thì S(1;n) không phải là số nguyên. Năm 1946, P. Erdos và I. Niven đã chứng minh được rằng chỉ có hữu hạn số nguyên dương n để tồn tại k sao cho S(k,n) là số nguyên.

Tốt hơn nữa, vào năm 2012, Yong-Gao Chen và Min Tang đã chứng minh được rằng nếu n>3 thì không có k sao cho S(k,n) là số nguyên.

Dưới đây là chứng minh của họ. Continue reading “On the elementary symmetric functions of 1, 1⁄2, …, 1⁄n”