[PDF] Top 20 Mathematical Excalibur, Volume 8, Number 2

... For theorem 2, let us list the elements of S n without repetition or omission in the n-th row of a table. (If S n is finite, then the row contains finitely many elements.) Now we can list the union of these sets ... See full document

... Problem 2. Let n ≥ 3 be an integer. In a conference there are n mathematicians. Every pair of mathematicians communicate in one of the n official languages of the conference. For any three different official ... See full document

... = 2, 3 時取最大值，k = 6 ， C 0 6 < C 1 6 < C 2 6 < C 3 6 > C 4 6 > C 5 6 > C 6 6 ， C m 6 在 m = 3 時最大值。）利用這個 關係，讀者可以証明，如果在容斥原 則的右邊，略去一個正項及它以後各 項，則式的左邊大於右邊，這是因為 x 對於右邊的貢獻非正，或者被略去的 貢獻非負。同理，如果在容斥原則的 右邊略去一個負項及它以後各項，則 ... See full document

... n 2 − b n 2 + a 0 ， 而虛部則等於 2a n b n + b 0 。 將以上的計算化成程序，得第 110 及 120 行。REC 和 IMC 分別是 c 0 的實值和虛值。 RE 和 IM 分別是 c n 的實值和虛值。 RE2 和 IM2 分別 是 c n ... See full document

... Problem 194. (Due to Achilleas Pavlos PORFYRIADIS, American College of Thessaloniki “Anatolia”, Thessaloniki, Greece) A circle with center O is internally tangent to two circles inside it, with centers O 1 and O 2 ... See full document

... Solution.: Independent solution by CHEUNG Cheuk Lun (S.T.F.A. Leung Kau Kui College). The positive integers are separated into two subsets with no common elements. S[r] ... See full document

... If the starting point lies within a distance of 1 from the origin, the subsequent points will get closer and closer to the origin.. If the intial point is more than a distance o[r] ... See full document

... On-line: http://www.math.ust.hk/excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone and fax numbers (if ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... Solution. LAM Cho Ho (CUHK Math Year 1). Take a circle of radius r so that all intersection points of the n lines are inside the circle and none of the n lines is tangent to the circle. Now each line intersects the ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... This year’s International Mathematical Olympiad (IMO) has been of considerable significance to Hong Kong. At the 1997 IMO held in Mar del Plata, Argentina, shortly after our official transfer of sovereignty, the ... See full document

... Problem 1, which was supposed to be a number theory problem, is more like an algebra problem (no prime numbers, no factorization of integers, merely algebraic manipulation and some induction). And finally of ... See full document

... Mathematical Excalibur, Vol. 19, No. 2, Sep. 14 – Oct. 14 Page 2 Problem ...• 2 be an integer. Consider a n ×n chessboard consisting of n 2 unit ...k 2 unit ... See full document

... polynomials in problem 1 and numbers in problem 2. Like vectors expressed in coordinates, the v i 's are objects that may take on different values at different positions. So functions corresponding to solutions of ... See full document

... Kin-Yin Li, Dept of Mathematics, Hong Kong Wniversiv of Science and Technology, Clear Water Bay, Kowloon. Eight students took part in a contest with eight problems[r] ... See full document

... x 2 , ..., x 1996 and let s i = x 1 + x 2 + ... + x i for i = 1, 2, ...1995 2 ), ..., [ 1994 1995 ,1) and the 1996 numbers {s 1 }, {s 2 }, ... See full document