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

标签:
传播不等式流年 |
分类: 奥赛 |





Inequalities Crux Mathematicorum.pdf
|

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

参考文献:sqing55.IMO练习曲
http://blog.sina.com.cn/s/blog_4c1131020102uxks.html#cmt_2914610
1998年伊朗数学奥林匹克决赛试题(英文版) -
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
(2)Inequalities
【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
.


http://blog.sina.com.cn/s/blog_4c1131020102uxks.html
2:熊昌进 IMO练中第8题不等式的证明
http://blog.sina.com.cn/s/blog_64e168d30102uzph.html
青年教师入行题50,51,52
i)
ii) Determine, reasonably, if there exist infinite pairs of integers