作者JohnMash (Paul)
看板Math
标题Re: [中学] 2011年青少年数学国际城市邀请赛
时间Sun Apr 17 13:10:51 2011
※ 引述《eric314 (爱困)》之铭言：
: 2 设a,b,c为实数,若
: n = (a+3c)/(a+2b+c) + (7a+6b+3c)/(a+b+2c) + (c-a)/(2a+b+c)
: 求 n 的最小值 ? Ans: 1+2根号3
a,b,c should be POSITIVE
[2 1 1]^{-1}=[ 3 -1 -1]
[1 2 1] [-1 3 -1] ×(1/4)
[1 1 2] [-1 -1 3]
Let
a=3u-v-w; b=-u+3v-w; c=-u-v+3w;
n=(vw^2+2uw^2-3uvw+2uv^2+3u^2v)/(uvw)
=w/u+2w/v-3+2v/w+3u/w
Let w/u=x, w/v=y
x+3/x+2(y+1/y)
x+3/x≧2√3
y+1/y≧2
n≧1+2√3
[I don't think it is a GOOD problem]
: 4 设有20段线段长分别为1,2,...,20的线段,从中任意挑选3线段长
: 来构成一个任意两边长得差至少为2至多为7的三角形,问
: 满足该条件的相异挑选方法共有多少种 ?
Let 1≦a<a+k<a+k+m≦20
and 2≦k≦7
and 2≦k+m≦7
and 2≦m≦7
and 2a+k>a+k+m (triangle inequality) i.e. a>m
--------------------------------------------------------
m=2, k=2,3,4,5, 3≦a≦18-k then a has 16-k choices
m=3, k=2,3,4, 4≦a≦17-k then a has 14-k ...
m=4, k=2,3, 5≦a≦16-k ...... 12-k
m=5, k=2, 6≦a≦15-k .... 10-k
N=4*16+3*14+2*12+10-4*2-3*3-2*4-5
=64+42+24+10-8-9-8-5=140-30=110
※ 编辑: JohnMash 来自: 112.104.208.207 (04/17 20:09)