12-08-09

給自己訓練一下數學吧~ 有空的話讓我做個數學100問 O_O~ Algebra, Ineq., Combinatorics, Number theorey, Geometry 各出20題然後自己解…… NT(number theorey) Q1. (n,m) denote G.C.D. of m,n. Prove (m,n)/n * nCm is an integer. solution lemma : for (m,n)=d, exists +ve integer a,b such that am+bn=d. (m,n)/n * nCm = (am+bn)/n * nCm = am/n * nCm + bn/n * nCm = a(n-1)C(m-1) + bnCm. ============================== 唉~上完呢堂IMO training真係#^%(# 佢講Extended Binomial Theorem講到我聽聽下將以前係pure maths學D Binomial thm忘記晒!! Extended Binomial Theorem: (1+x)^n = 1 + nx + (n r) x^2 + ... where (n r) = [n(n-1)(n-2)...(n-r+1)/r!] for all reals. Examples : How many natrual numbers, less than 100000, have their digit sum equal to 17？ Solution: <=> (1+x+x^2+...x^9) ^5, the coeff. of x^17. <=> (1-x^10)^5(1-x)^-5 =..... => 4435. ============================== 升到#6了 29 hours agowmfchris has risen 1 rank, now placed #6 overall (Taiko). 30 hours agowmfchris has risen 1 rank, now placed #7 overall (Taiko). 30 hours agowmfchris has risen 1 rank, now placed #8 overall (Taiko). 35 hours agowmfchris has lost first place on Home Made Kazoku - Shonen Heart [Hard] (Taiko) 36 hours agowmfchris has risen 20 ranks, now placed #657 overall (CatchTheBeat). 36 hours agowmfchris has risen 2 ranks, now placed #7 overall (Taiko). 36 hours agowmfchris has risen 1 rank, now placed #9 overall (Taiko). 保持著"不失#1超過24小時"紀錄。 ============================== 我大概被楠和紫欣痛毆N次了~ 拖稿那麼久……不過我還是15/8以後才有空啊。