Mathematical Olympiad number theory 3 In a series of numbers: 1 / 1,1 / 2,2 / 2,1 / 2,1 / 3,2 / 3,3 / 3,2 / 3,1 / 3,1 / 4,2 / 4,3 / 4,4 / 4,3 / 4,2 / 4,1 / 4,1 / 5,..., which fraction is 7 / 10, and what is the 400th fraction?

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• 1. There are 16 pieces in a row on the table. Party A and Party B take turns to take at least one piece at a time and at most two pieces at a time. Party A takes the first and who gets the last one is the winner. Q: who can win? What strategy should be adopted to win?
• 2. There is a round table on which Xiaoming and Xiaofang play chess. The rules of the game are: each person gets one chess piece at a time and puts it on the table, and the two people take turns to place it. The chess pieces are not allowed to be covered or overlapped. When there is no more place to put the chess pieces on the table, the game ends. At this time, the last person to place the chess pieces wins, How to place the pieces?
• 3. Natural numbers a and B are not multiples, and their product has nine divisors. The difference between their least common multiple and the greatest common divisor is 224. What is the difference between the two natural numbers?
• 4. Fill in the following numbers: 3, (), 6,90,9,80, (), () Fill in the following numbers: 3, (), 6,90,9,80, (), ()
• 5. Observe the following triangle number matrix to find the rule: one 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .................. As shown in the figure above, the number 21 from left to right in line 22 from top to bottom is (), and 2010 is the number () in line () one 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .................. As shown in the figure above, the number 21 from left to right in line 22 from top to bottom is (), and 2010 is the number () in line ()
• 6. Three numbers to find the rule （1）、0,1,3,8,21,（ ）,144 （2）、0,1,4,15,56,（ ） （3）、1,3,6,8,16,18,( ),( ),76,78 And the trouble narrates seeks the rule
• 7. The number of triangles to find the law. Urgent! Lead a line segment from a vertex of a triangle to the opposite side. Lead 0, it is a triangle; lead 1, it will count 3 triangles; lead 2, it will count 6 triangles; lead 3, it will count 10 triangles Quoting n, you can count out () triangles
• 8. Four squares aba1b1, bcb1c1, cdc1d1 and dad1a1 are made out of the four sides AB, BC, CD and Da of the quadrilateral ABCD. The centers of the four squares are m, N, P and Q respectively, connecting MP and NQ. It is proved that MP = NQ and MP is perpendicular to NQ
• 9. It is known that a, B and C are three non negative numbers, and satisfy 3A + 2B + C = 5, 2A + b-3c = 1. Let m = 3A + b-7c, find the maximum and minimum of M Please give me a detailed process
• 10. The rules are: the first number is 1, the second number is 2, and the third number is 1. Generally, write a line of 1, and then insert K 2 (k = 1, 2, 3,...) between the K 1 and K + 1 1 1 Question (1) is the number 2005 1 or 2? (2) What is the sum of the first 2005 numbers?
• 11. There are string numbers: 1,2,4,7,11,16,22. How many of the first 50 numbers in this string are divided by 3 to 1?
• 12. Find the rule: 1,2,4, () 11, () () what number to fill in brackets
• 13. Depressed ah, can't do the number of primary school to find the law of the problem! There is a certain rule for each group of numbers below. What is the rule? 1.2,4,12,16,30 2.1,3,9,15,27 3.5,10,20,25,45 4.3,9,15,21,33
• 14. There is a big circle in the middle, 10 in it, a small circle in the top, 2 in it, two small circles in the bottom, 3 on the left and 5 on the right. There is a big circle in the middle, 30 in it, a small circle in the top, 3 in it, two small circles in the bottom, 5 on the left and 6 on the right. There is a big circle in the middle, 64 in it, 4 in it, two small circles in the bottom and 6 on the left, The last one on the right is 8. The small circle on the top is 5 inside, and the two small circles on the bottom are 9 on the left and 4 on the right. How about the number of the big circle in the middle of the last one
• 15. How to find rules and fill in numbers in primary school (super simple) 1,2,3,10,17,（ ）,（ ）
• 16. Primary school to find a rule to fill in the number, simple 2 / 5 4 / 5 1.2 one and three fifths () () 1 1/4 1/9 1/16 1/25 （ ） （ ）
• 17. Primary school to find the rule to fill in the number 6,1,8,3,10,512,7（）（） 15,16,13,19,11,22（）（）7 1/1.3/4,2/3,5/8,3/5,7/12()() 1,3,8,16,27,41()()
• 18. 1： 3-3 () + 1 () - 2 () + 3 () - 1 () 2： 4-1 () + 0 () + 2 () - 1 () + 3 () 3： 2 + 3 () - 1 () - 2 () - 1 () + 3 () 4： 5-1 () - 2 () + 2 () + 1 () - 1 () Ask () what to fill in and what rules to follow
• 19. Fill in the numbers regularly in grade one 3.4.9.9.15.14. (). 19. ()
• 20. Fill in the numbers in the first grade of primary school