Which way did the bicycle go (medium)

14. Interestingly, some simple conclusions have not been found in the history of mathematics for centuries. It was not until 1977 that Paul ERD ipvs and George szekeres found that, except for the two sides of 1, any two numbers in the same row of the Yang Hui triangle had a public factor. Prove this conclusion.

Answer: just note that a multiplied by a number smaller than B can become a multiple of B, which indicates that a and B must have a public factor. Set 0
 
C (n, j) · C (J, I)
= N! /(J! (N-j )!) · J! /(I! (J-I )!)
= N! /(I! (N-j )! (J-I )!)
= N! /(I! (N-I )!) · (N-I )! /(J-I )! (N-j )!)
= C (n, I) · C (n-I, j-I)

 
5 times of 111 is 10, 37 times of 3 is 100, and 25 times of 4 is. Can we find a multiple of N for any positive integer N, which is composed of digits 0 and 1?

Answer: Yes. Consider Series 1, 11,111,111 1 ,.... There are only N possibilities for dividing them by the remainder of N. Therefore, there are two items in the first n + 1, and they are the same by the remainder of N. The difference between the two conditions is met.

 
16. You may often notice the interesting fact that 111 can be divided by three. Is there an infinite number of positive integers where n satisfies, and the N-digit numbers composed of N 1 can be divisible by N?

Answer: Yes. We only need to prove that, if the N digits of N numbers composed of N 1 can be divisible by N, the 3N digits of the 3N one can be divisible by 3n. This is because 11 .. 11 11 .. 11 11 .. 11 can be written as 11 .. 11*1 00 .. 01 00 .. 01, where the former contains factor n, and the latter obviously contains Factor 3.

 
17. can I find a multiple of N for any positive integer N, which contains all numbers from 0 to 9?

Answer: Yes. Assume that N is a D-digit number, then there must be a number between 1234567890 · 10 ^ d + 1 and 1234567890 · 10 ^ d + n that is a multiple of N, which obviously meets the requirements.

 
18. for any positive integer set a, so that S is the set of all and which may be obtained by adding the numbers in, so that D is a set of all the differences that may be obtained by the subtraction of the numbers in. For example, if a = {1, 2, 4}, S = {2, 3, 4, 5, 6, 8}, D = {-3,-2, -1, 0, 1, 2, 3 }. Proof or overturn: the number of elements in D cannot be less than the number of elements in S.

Answer: This is wrong. Currently, the minimum inverse examples are known to be {1, 3, 4, 5, 8, 12, 13, 15}. These 8 numbers can produce 26 kinds of sum, but only 25 kinds of difference.

 
19. polynomial p (x) = (1/2) x ^ 2-(1/2) x + 2 satisfies P (1) = 2, P (2) = 3, P (3) = 5. Can I findIntegerPolynomial Q (x), so that Q (1) = 2, q (2) = 3, Q (3) = 5?

Answer: No. In fact, the integer polynomials that only meet Q (1) = 2 and Q (3) = 5 do not exist. Suppose Q (x) = A0 + A1 · x + A2 · x ^ 2 +... + an · x ^ N, then
 
3 = 5-2 = Q (3)-Q (1) = (3-1) A1 + (3 ^ 2-1) A2 +... + (3 ^ N-1).
 
Because 3 ^ k-1 is always an even number, so the right side of the equation must be an even number, and it cannot be 3. There is a conflict.

 
20. assume that p (x) is an eight polynomial and P (1) = 1, P (2) = 1/2, P (3) = 1/3 ,..., P (9) = 1/9. Evaluate P (10 ).

Answer: The conditions show that 1, 2,..., 9 is the 9 root of Polynomial x · P (x)-1. Therefore, X · P (x)-1 = C (x-1) (X-2) (X-3)... (X-9 ). We can see that-1 =-C · 9! , So c = 1/9! . Therefore, 10 · P (10)-1 = 9! /9! = 1, so P (10) = 1/5.

 
21. Write Yang Hui's triangle into a square matrix:

1 1 1 1...
1 2 3 4 5...
1 3 6 10 15...
1 4 10 20 35...
1 5 15 35 70...
...
...

Proof: For any positive integer N, the matrix consists of n columns of the First n rows of the matrix, the total determinant is 1.

Answer: sum up n. When n = 1, it is obviously true. Consider n columns in the first n rows of the square matrix. If each row is subtracted from the previous row, it becomes:
 
1 1 1 1...
0 1 2 3 4...
0 1 3 6 10...
0 1 4 10 20...
0 1 5 15 35...
...
...
 
Subtract the previous column from each column:
 
1 0 0 0 0...
0 1 1 1...
0 1 2 3 4...
0 1 3 6 10...
0 1 4 10 20...
...
...
 
Obviously, its determinant is the same as that of N-1 order.

 
22. When shuffling a machine, the order of cards is always disrupted in the same way. Set

A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Put it in, use the machine to wash the cards twice in a row, and then change the order

10, 9, Q, 8, K, 3, 4, A, 5, J, 6, 2, 7

Find the order of the machine after the first shuffling.

Answer: We can regard this card washing machine as a replacement σ, then σ ^ 2 is
 
1 → 8 → 4 → 7 → 13 → 5 → 9 → 2 → 12 → 3 → 6 → 11 → 10 → 1
 
σ ^ 2 cannot be divided into several non-Intersecting loops, so σ cannot have multiple loops. However, this indicates that all cards will return to the original position after 13 consecutive shuffling. Therefore, a card is equivalent to (σ ^ 2) ^ 7, that is
 
1 → 2 → 8 → 12 → 4 → 3 → 7 → 6 → 13 → 11 → 5 → 10 → 9 → 1
 
So the order is
 
9, A, 4, Q, J, 7, 3, 2, 10, 5, K, 8, 6