ลองทำดูมั้ยครับ..

[Mathopolis]

1. A while ago in a laboratory in an african country, a dangerous bacteria named "Ishangololo" was made by an accident. After maturity, this bacteria splits into two bacterias like itself. The age of maturity might not be the same in all bacterias. We say that the bacteria A is a child of the bacteria B if A has came into existence by the division of the bacteria B of by the division of one of B's children. Right now, we have 3k bacterias in the lab. Prove that there is a bacteria whose number of children among the current bacterias is a number greater or equal than k and smaller or equal than 2k.

2. Around a circle, we have written n numbers each either 0 or 1 and one of these n numbers is underlined. In each round, we can do one of the below :
i) Remove the line below the underlined number and underline one of the numbers next to it.
ii) Change the underlined number from 0 to 1 and vice versa.
Find the least number of rounds needed so that we can change any first condition to another.

3. A 8x8 square on the plane divided into 64 squares 1x1must be covered by 64 black and 64 white isosceles triangles (each square must be covered by two triangles). A covering is "fine" if any two neighboring (having a common side) triangles are of different colors. How many different "fine" colorings are there?

4. A journalist is looking for a person named Z at a meeting of N persons. He has been told that Z knows all the other people at the meeting but none of them knows Z. The journalist may ask any person X : 'Do you know that man?' about any Y.
a) Can he be certain to find Z by asking less than N questions?
b) What minimal number of questions are needed to find Z? (Prove that a lesser number won't work.)

5. A rectangle axb (a>b) is cut into right triangles so that any common side of two triangles is a leg (cathetus) for one of them and a hypotenuse for the other. (Every two triangles have a common side or a common vertex or no common points.) Prove that the ratio a/b is not less than 2.

6. 8 students were asked to solve 8 problems. (the same ones)
a) Each problem was solved by 5 students. Prove that one can find two students so that each problem was solved by at least one of them.
b) If each problem was solved by 4 students, then it is possible that no such a pair of students exists. Prove this.

7. The integers 1,2, ... , n each appear n times in an nxn matrix. Show that there is a row or column containing at least n1/2 distinct integers.

8. There is a race car on a circular racetrack of n miles. There are fuel stations around the track, such that the total amount of fuel at all the stations is just enough to travel n miles. Show that there is a point on the track at which the race car may start, such that it can complete a circuit by picking up fuel along the way.

9. Define a sequence of strings as follows: Let R0 = 1 , and let Rn be the string obtained by replacing each instance of 'i' in Rn-1 with '123..i' and finally adding 'n+1' at the end. So for instance, R1 = 12 , R2 = 1123 , R3 = 11121234 . Show that for n>0 , if we write Rn down and Rn backwards under that, then each column will contain exactly one '1' .

10. Let m and n be positive integers. Let x1 , x2, ... , xn be non negative integers, whose average is less than or equal to m+1, and let y1 , y2, ... , yn be non negative integers, whose average is less than or equal to n+1. Show that some non empty subset of the xi's has sum equal to some non empty subset of the yj's.

11. 8 singers take part in an art festival. The organizer wants to plan m concerts. 4 singers go on stage in every concert. Restrict that the times of which every two singers go on stage in a concert are all the same. What is the minimal of m? (Make a design.)

12. Between 2n-1 positive integers, prove that we can choose n of them in which the sum of these n integers is divisible by n.

13. On the island of Camelot, live 13 gray, 15 brown and 17 crimson chameleons. If two chameleons of different colors meet, they both simultaneously change color to the third color. Is it possible that they will eventually all be the same color?

14. The numbers x1 , x2 , ... , xn are such that x1 + x2 + ... + xn = 0 and x12 + x22 + ... + xn2 = 1 . Prove that there are two numbers among them whose product is no greater than -1/n.

15. We say that some positive integer m "covers" the number 1998, if 1, 9, 9, 8 appear in this order as digits of m. (For instance 1998 is covered by 215993698 but not by 213326798.) Let k(n) be the number of positive integers that cover 1998 and have exactly n digits (n>4), all different from 0. What is the remainder of k(n) in division by 8?

16. The cube S with edge 2 consists of 8 unit cubes. We will call the cube S with one unit cube removed, a "piece". The cube T with edge length 2n consists of (2n)3 unit cubes. Prove that if one unit cube is removed from T, then the remaining solid can be built with pieces.

17. Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled.
a) Prove that Peter can always transfer his money into two accounts.
b) Can Peter always transfer his money into one account?

18. On a straight line, n red points and n blue points (not necessarily distinct) are marked arbitrarily. Prove that the sum of the distances between every two points of different colors is not less than the sum of the distances between every two points of the same color.

19. A 7x7 square is made up of 16 '1x3' tiles and 1 '1x1' tile. Prove that the 1x1 tile lies either at the center of the square or adjoins one of its boundaries.

20. At a round table are 1994 girls playing a deck of n cards. Initially, one girl holds all of the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is holding at most one card.
a) Prove that if n > 1993, then the game cannot end.
b) Prove that if n < 1994, then the game must end.

[tatari/nightmare]

เอามาจากไหมครับเนี่ย

[putmusic]

อ่านไม่รู้เรื่องเลยครับ

[konkoonJAi]

12. Between 2n-1 positive integers, prove that we can choose n of them in which the sum of these n integers is divisible by n.
วิธีทำ พิจารณาการเลือก n จำนวน จากจำนวน 2n-1 ผลบวกที่เป็นไปได้น้อยที่สุดคือ $\sum_{i = 1}^{n}i$ และผลบวกที่เป็นไปได้มากที่สุดคือ $\sum_{i = 1}^{2n-1}i-\sum_{i = 1}^{n-1}i$ ดังนั้นผลบวกที่เป็นไปได้ทั้งหมดคือ $\sum_{i = 1}^{2n-1}i-\sum_{i = 1}^{n-1}i-\sum_{i = 1}^{n}i+1=\frac{(2n-1)(2n)-(n-1)n-n(n+1)}{2}=n(n-1)+1 > n$ ให้ n(n-1)+1 แทนจำนวนนก ให้รังแทนคลาสของเศษที่เกิดจากการหารจำนวนใดๆ ด้วย $n$ ซึ่งมีทั้งหมด $n$ รัง โดยหลักการรังนกพิราบแบบเข้มจะได้ว่า มี n จำนวนที่มีเศษจากการหารด้วย $n$ เท่ากัน

[nooonuii]

มันต้องเป็นผลบวกของทั้ง $n$ ตัวไม่ใช่เหรอครับ

[konkoonJAi]

แก้ไขแล้วนะคะ รบกวนพี่ nooonuii ช่วยดูอีกทีด้วยค่ะ

[nooonuii]

ความหมายของโจทย์คือเราต้องเลือก $a_1,...,a_n$ ซึ่งทำให้ $n|a_1+\cdots +a_n$ ไม่ใช่เหรอครับ จริงๆก็ทำมาเกือบเสร็จแล้วนะครับ

[konkoonJAi]

ขอบคุณมาก ๆนะคะ