作者cuttlefish (睡觉真是一种幸福呀~ Ｉ
看板IMO_Taiwan
标题Re: 从云飘飘转来的...放在这比较多人看~
时间Thu Sep 18 10:39:22 2003
※ 引述《cuttlefish (睡觉真是一种幸福呀~ Ｉ》之铭言：
第四题好像可以用这个不等式:
\
if a 为正实数
sqrt(a)>=2a/(a+1)
\
ps我只会这题啦~
有时间..写详细一点好了(我想大家应该早就会了....)
sqrt(a)>=2a/(a+1) 等号成立在 a=0 or 1
sqrt[x/(y+z)]>=2x/(x+y+z)
sqrt[y/(x+z)]>=2y/(x+y+z)
sqrt[z/(x+y)]>=2z/(x+y+z)
相加即可
注意到等号不能全成立
得证
※ 引述《chaogold (dchaodx)》之铭言：
: ※ 引述《myflame (龙队 红队 太阳队)》之铭言：
: : 感谢chaogold提供题目
: : 1. In triangle ABC prove that a + b + c < r_1 + r_2 + r_3 + min {r_1,r_2,r_3}
: : where r_1 is the exradius opposite a, r_2 is the exradius opposite b and r_3
: : is the exradius opposite c.
: : 2. How can you arrange numbers from 1 to 2003 in a row so that avg. of any
: : two numbers doesn't lie between them? E.g. 2003...1002...1 is invalid as
: : (1+2003)/2=1002
: : 3.Prove that there are infinitely many primes that can be written as the sum
: : of a prime and a power of two.
: : 4. [x/(y+z)]^(1/2) + [y/(x+z)]^(1/2) + [z/(y+x)]^(1/2) > 2
: 第二题学长的解法太妙了！
: 我只能说妙..
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1F：→ JGU:咳~你没事干麻翘普物跑来计中电别人=.= 推 61.230.34.82 09/18
2F：→ cuttlefish:翘普物来计中是要看某消息(魔术社的).... 推140.112.249.199 09/20
3F：→ cuttlefish:而且我才电不到人呢~ 推140.112.249.199 09/20