加载中…
个人资料
sqing55
sqing55
  • 博客等级:
  • 博客积分:0
  • 博客访问:3,296,084
  • 关注人气:2,237
  • 获赠金笔:0支
  • 赠出金笔:0支
  • 荣誉徽章:
正文 字体大小:

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

(2014-07-16 15:54:51)
标签:

传播

不等式

流年

分类: 奥赛

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

6h323869p1414457655

                IMOT 2011IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)                                       1998伊朗数学奥林匹克决赛试题

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

Inequalities Crux Mathematicorum.pdf

                                                          problem  night 1

 http://www.artofproblemsolving.com/community/c6h599319p3557298

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

                       Netherlands IMO Team Selection Tests 2014
                  IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)                 
                                                                 Dutch IMO TST I Problem 3
                     Spain Spain Mathematical Olympiad 2014                   IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

                         hoangthailelqd --arqady不等式           

        IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

                      mathwizarddude不等式

               IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum) 
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
 
 

IMO练习曲部分试题解答1

这段时间忙于装修,很少有连贯的时间来做题。

今天原计划去木工厂看家具,临出门了反悔。从十点多做到现在,解决了宋庆老师帖子:IMO练习曲(附:不等式选摘:Inequalities proposed in Crux Mathematicorum)里面十多道不等式,感觉有点饿了。

学习宋老师数学圈子里面的优良传统,选了三题的解答贴出来。

http://wuling.name/wp-content/uploads/2014/07/%E6%8D%95%E8%8E%B7.jpg

http://wuling.name/archives/2619
                                                           张云华:证等价
IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
 
                                                 张云华:证一个三元代数不等式

 

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

                                                      IMO练习曲
 
                                                          648 宋庆 IMO练习曲

参考文献     

1:宋庆 IMO练习曲

http://blog.sina.com.cn/s/blog_4c1131020102uxks.html

2熊昌进 2014西班牙奥林匹克第2题的证明

 http://blog.sina.com.cn/s/blog_64e168d30102uy07.html

 

                  不等式选摘:Inequalities proposed in Crux Mathematicorum   reny

【1528*】Let http://data.artofproblemsolving.com/images/latex/6/4/6/64621ec329cc65c4cba6449f948109e80a6b2630.gif . Then the equation of the tangent line of http://data.artofproblemsolving.com/images/latex/6/5/4/65405422ff71ebf2db437dbd89a41355f4f19183.gif at http://data.artofproblemsolving.com/images/latex/1/8/3/1837e427cb0198f25234c1c37ed293f9d15fa96a.gif is given by http://data.artofproblemsolving.com/images/latex/5/5/c/55c7395f7199c1bb1eef14d15f45e07279a83817.gif . To verify this, note that the slope of the tangent line at http://data.artofproblemsolving.com/images/latex/8/e/4/8e4168bab9f6f7f976a26088a6b800a277387daa.gif is http://data.artofproblemsolving.com/images/latex/1/c/9/1c9787560f88e6319caebde345278ed4427af52b.gif , where http://data.artofproblemsolving.com/images/latex/e/c/d/ecde1a39d56bdcb8f7b29b5b53c0890e46a026cd.gif . As http://data.artofproblemsolving.com/images/latex/1/a/d/1adee0f1e46aabe3f32c9f9b8ae15062bdc6a69b.gif , we get the desired equation.

Now, consider the function http://data.artofproblemsolving.com/images/latex/6/9/0/690b10ddec504bf17ab3f68e2ed29c1f99332186.gif for http://data.artofproblemsolving.com/images/latex/0/4/a/04ab9eb41d7f0b293f72cf64d0faed0f3c76b5a9.gif . Then http://data.artofproblemsolving.com/images/latex/1/b/f/1bfe1e23d4b8da9583c7b3ff78bc9b4637f6b476.gif . Note that the equation http://data.artofproblemsolving.com/images/latex/d/f/a/dfa7057011f5b8fb9fae43803cd6c9d08e80e267.gif has exactly one positive real root http://data.artofproblemsolving.com/images/latex/4/7/a/47ade823cf84a8e4343c2c0cd70ab91c00d7cbd8.gif by Descartes' Rule of Signs. Thus, http://data.artofproblemsolving.com/images/latex/3/8/3/38376e123e13462b554eaa3482588b55675030d8.gif for http://data.artofproblemsolving.com/images/latex/8/a/5/8a52cce7db4f30cd1130daf51c3362843147468c.gif .

It follows that if http://data.artofproblemsolving.com/images/latex/5/4/0/5402bf18d0f112eaf38a6e3db9e115a3901d9415.gif and http://data.artofproblemsolving.com/images/latex/c/7/0/c701506b30199fb62872e07265092ff4dc4b9f38.gif , then
http://data.artofproblemsolving.com/images/latex/9/0/a/90aead956b158c7e0c1e2b16f6086579ba3280c0.gif
Now, suppose that at least one of http://data.artofproblemsolving.com/images/latex/3/d/c/3dcda24350a7219c75a34cb4f0079978d4b63e95.gif is less than http://data.artofproblemsolving.com/images/latex/e/f/b/efbda784ad565c1c5201fdc948a570d0426bc6e6.gif . WLOG, let http://data.artofproblemsolving.com/images/latex/c/5/e/c5e91f38100034476b86a076dbc813f93b7e5549.gif be fixed. Set http://data.artofproblemsolving.com/images/latex/5/6/8/568d9d61d0303e6157688a80a7190fdc75260f32.gif , so http://data.artofproblemsolving.com/images/latex/2/c/6/2c6bd974a0ea16da6afbe3b8461f244c50e77514.gif .
Since http://data.artofproblemsolving.com/images/latex/e/1/a/e1a103ce7f11376ecb1efabd2d38c21517df3ec7.gif , the point of inflection of http://data.artofproblemsolving.com/images/latex/6/5/4/65405422ff71ebf2db437dbd89a41355f4f19183.gif is http://data.artofproblemsolving.com/images/latex/0/1/b/01b7aa2ec27a082adc3f2b1f284f1bbd155302fd.gif , where http://data.artofproblemsolving.com/images/latex/2/9/b/29bdf45f25da3e8cd2c1d3c106649220f1470eec.gif , which is approximately http://data.artofproblemsolving.com/images/latex/6/9/5/69527126a90b331ca803eefb8c9e3416cec0452a.gif . Also, the tangent line at http://data.artofproblemsolving.com/images/latex/0/d/9/0d9746b78e2f432fea0841025dbbccc26614859d.gif is given by http://data.artofproblemsolving.com/images/latex/6/b/4/6b4ecd4fa831695355faa433aabc71f3e40ac8f8.gif . Since http://data.artofproblemsolving.com/images/latex/9/8/c/98c0e729007be10990b30a17841de49def4a9b2e.gif , the tangent line at http://data.artofproblemsolving.com/images/latex/0/d/9/0d9746b78e2f432fea0841025dbbccc26614859d.gif is above the curve. Furthermore, one can verify that the tangent line at http://data.artofproblemsolving.com/images/latex/b/a/b/bab0ee9429fd55f221ccfc7d4158cdce17a2b86c.gif passes through the origin. Since http://data.artofproblemsolving.com/images/latex/9/7/6/9763def0f8813c51179d00acf7715593efe1b9a4.gif , the tangent line at http://data.artofproblemsolving.com/images/latex/0/d/9/0d9746b78e2f432fea0841025dbbccc26614859d.gif is still above http://data.artofproblemsolving.com/images/latex/6/5/4/65405422ff71ebf2db437dbd89a41355f4f19183.gif when http://data.artofproblemsolving.com/images/latex/8/b/d/8bdf21367eee06097384c37b0448375f07f950f0.gif . Therefore, we have http://data.artofproblemsolving.com/images/latex/2/3/7/237231f16b251324bc81c53f1bfc8e3c8dedabb2.gif . It follows that
http://data.artofproblemsolving.com/images/latex/e/6/e/e6e21ede9c5dad1cbf0442c45d956634493c081e.gif
It suffices to show that http://data.artofproblemsolving.com/images/latex/b/8/1/b8145a9b59751456b5826f965783b11def99819b.gif , where http://data.artofproblemsolving.com/images/latex/d/d/d/dddf4bbf97360b3a7599162b8d5bc1f2e5a8b3d1.gif . We have
http://data.artofproblemsolving.com/images/latex/d/a/e/dae182dbbe547fb23acac889c40652810e9ec2f2.gif
and because http://data.artofproblemsolving.com/images/latex/4/e/5/4e5b1b5611c3bd20118d1a5e8b627bed5b020402.gif has a root between http://data.artofproblemsolving.com/images/latex/7/7/d/77de68daecd823babbb58edb1c8e14d7106e83bb.gif and http://data.artofproblemsolving.com/images/latex/1/b/6/1b6453892473a467d07372d45eb05abc2031647a.gif , with http://data.artofproblemsolving.com/images/latex/e/d/1/ed1265a277a25be259cd4e8bc838721dee4873c2.gif and http://data.artofproblemsolving.com/images/latex/a/8/2/a82501346841e30ee6c23e06f771e616a8a43046.gif , we have http://data.artofproblemsolving.com/images/latex/b/9/9/b994e2cb89f49757479fcb579ad70448fabda237.gif for http://data.artofproblemsolving.com/images/latex/3/c/7/3c7bd3b71d38a28737b9124e3109bff4341fc60a.gif .
Therefore, the inequality is true and equality holds iff http://data.artofproblemsolving.com/images/latex/0/b/1/0b1a47c8c914f09803cbd5a6b103d7be73ed5f2b.gif .
 一个三元分式的最小值  reny

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)
                            同一种方法解两题  一个三元不等式  reny IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

 

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

 

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

 

 

 

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)

 

 

IMO练习曲(附:不等式选摘:InequalitiesproposedinCruxMathematicorum)


1. 已知 http://data.artofproblemsolving.com/images/latex/3/5/e/35e7ccd368191887013a91477542e5bd4c556889.gif                                         

                                                                                                                              problem 1 night 4      

 

2. 已知 http://data.artofproblemsolving.com/images/latex/e/f/2/ef26eb473b1f8fd1c938168fc847c2a08134f0c1.gif 等价于

                http://data.artofproblemsolving.com/images/latex/7/0/b/70b782e0cfa12b7b7a08b0e3204eb167557ea5ba.gif                                                               problem 7 night 3

3.已知 http://data.artofproblemsolving.com/images/latex/6/5/3/65337fffeff29f536470c5190cb7260be07436f2.gif 求证: 

               http://data.artofproblemsolving.com/images/latex/7/3/4/7341993aa15bf8a4f19ba09c94a9bdd97e3fb86c.gif                                  problem 4 night 10

                                                                           1998伊朗数学奥林匹克决赛试题

4. 已知http://data.artofproblemsolving.com/images/latex/1/2/d/12d2d1041d4c8eb72d868b13434c866def010f5c.gif

                                                                                                                              problem 7 night 8

5.已知http://data.artofproblemsolving.com/images/latex/7/e/d/7edf37e8eed95100a31170bc21e34e4f64f72663.gif的正数 ,求证 :
               http://data.artofproblemsolving.com/images/latex/a/a/1/aa121a7d67a8aee4af284d05626afda8ecb022d3.gif  
                                                                                                                              problem 1 night 5

6.已知http://data.artofproblemsolving.com/images/latex/a/0/b/a0b3afd6a86cf8fd1bedb3a573c524e68becf251.gif 两两不同, 均不为零,且http://data.artofproblemsolving.com/images/latex/1/2/1/121b9b6e283888a75381e9b3469f47020ef93efd.gif 求证: 

             http://data.artofproblemsolving.com/images/latex/2/2/c/22ce1fb93b7f857abcbfef94f7df11b1298c19bd.gif                  problem 2 night 5

7.已知 http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 为三角形三边长. 求证 :
a.  http://data.artofproblemsolving.com/images/latex/e/c/5/ec5335c8fae1dbfd152caba552f51b6dbb504b82.gif 为一三角形三边 长
b. http://data.artofproblemsolving.com/images/latex/9/d/b/9dbd392c0d12ba00174c5e2da82420f7374e3f50.gif            problem 1 night 3

8.已知 http://data.artofproblemsolving.com/images/latex/4/a/8/4a81e6858f701ce5073514103a90423eaf7028f8.gif  的最小值

                                                                                                                               problem 7 night  

9.已知 http://data.artofproblemsolving.com/images/latex/3/b/e/3be7a55cba1b01507e4a72daaff3ecdd9539f1b2.gif   

                                                

                                                                                                                                problem 2 night 1

10.已知 http://data.artofproblemsolving.com/images/latex/b/c/3/bc32aae2c2ed109e8c9ed84a428c8e85b7963cbf.gif 的任意一个排列.求证:

    http://data.artofproblemsolving.com/images/latex/0/a/c/0ac8350dffa8798fea5ad22834644b0f7acdc7ac.gif                                           problem 6 night 5

11.求证http://data.artofproblemsolving.com/images/latex/2/6/b/26b808893af1ddee7377b540816a50afeb3b4481.gif                                                                 problem 6 night 3
12.求证:http://data.artofproblemsolving.com/images/latex/2/9/0/2907a9c0e83ab59641cba7a96b52a880fcb4c6ab.gif                                                        problem 3 night 11
13.已知 http://data.artofproblemsolving.com/images/latex/4/7/c/47c1b75c6bbfa3a80baadc253599bda9d6e44efd.gif 求证:

http://data.artofproblemsolving.com/images/latex/c/1/0/c10cbf9de058bcb9dea0f27c970110656ba53a9c.gif                                             problem  night 1
                                                                                                                                     Dutch IMO TST I Problem 3

                                               Spain Spain Mathematical Olympiad 2014
 
                  1.   已知 http://data.artofproblemsolving.com/images/latex/9/9/c/99c503d72106bf2cec158c674d95d91ef022ed4a.gif 是一个有理数.                                                                                                                                            OME 2014 2



                                                            hoangthailelqd --arqady不等式
             已知http://data.artofproblemsolving.com/images/latex/5/8/d/58d109df795ab880c2de1e58198903d29a848343.gif.

                                                             mathwizarddude不等式

                                                            http://data.artofproblemsolving.com/images/latex/c/6/2/c6264dc1325304f7106a8308d73861fa5aea1c76.gif  .
 
 

0

阅读 收藏 喜欢 打印举报/Report
  

新浪BLOG意见反馈留言板 欢迎批评指正

新浪简介 | About Sina | 广告服务 | 联系我们 | 招聘信息 | 网站律师 | SINA English | 产品答疑

新浪公司 版权所有