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Five olympiad math problems
1. There is a class of numbers, all multiples of 7 and 8, and adding 9 is a prime number. What is the smallest number in this class?

Solution: The minimum number of this class is: 7*8*4+9=233.

2. Stack 2 100 cubes with side length of 1 cm into a cuboid. Its height is 1 decimeter, and its length and width are greater than its height. Its length is15cm and its width is14cm.

3. Three students, A, B and C, go to buy books together. The number of books they buy is double digits, with A buying the most and C buying the least. Knowing that the sum of these books is even, and their product is 3960, how many books can B buy at most?

Solution: 3960 = 2 * 2 * 3 * 5 * 6 *11.

=22

*

10

*

18

That is, B can buy 18 books at most.

The sum of two numbers is 73. If you tick off one of the large numbers, you will get a decimal. What is this big number?

Solution: Let a large number be10A+B.

, then the decimal is a.

, and then:

10a+b+a=73

1 1a+b=73

a=6

b=7

That is, this large number is 67.

5. Decompose 1000 into the sum of two natural numbers, one is a multiple of 7 and the other is a multiple of 1 1. If one of these two natural numbers is required to be as large as possible and the other is as small as possible, what are these two numbers?

Solution: 1000=2 1+979.

That is, these two numbers are 2 1 and 979.

6. The sum of the first 29 natural numbers plus 200 1 equals the sum of the other 29 natural numbers. What is the smallest of these 29 natural numbers?

Solution: (1+29) * 29/2+2001= 2436.

Let the smallest be a.

, the maximum quantity is (a+28)

, and then:

(a+a+28)*29/2=2436

2a+28= 168

a=70

That is, the smallest number is 70.