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第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答

(2014-03-12 12:45:45)
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第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
1.第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
1.(ThirdTimeLucky )Lemma 1: Suppose http://data.artofproblemsolving.com/images/latex/0/0/c/00c1bff9de8d4fc009abbfd696151a6d58535852.gif meets http://data.artofproblemsolving.com/images/latex/0/f/4/0f4d56d1e20778bf2e1052ecb3219509238fb660.gif at http://data.artofproblemsolving.com/images/latex/a/7/e/a7ee38bb7be4fc44198cb2685d9601dcf2b9f569.gif . Then http://data.artofproblemsolving.com/images/latex/9/d/4/9d4fc26a50bdeb6214f335053db2509f2e6393c1.gif are collinear.

Proof: It suffices to show http://data.artofproblemsolving.com/images/latex/8/3/5/835aa494b205bb7a512a2f23357cb52ef2fe9f3c.gif is cyclic and http://data.artofproblemsolving.com/images/latex/b/8/6/b86bd135c70e92d6b2051a988af8d8fd3b21cb67.gif isosceles and thus making http://data.artofproblemsolving.com/images/latex/b/7/8/b7801d3332d72be00bb742186513047b1a0d3ac0.gif the perpendicular bisector of http://data.artofproblemsolving.com/images/latex/5/b/3/5b3e3191fd7659c5e63ab6620a903048b48bcb56.gif . Let http://data.artofproblemsolving.com/images/latex/d/5/3/d53599f21923e6269c18fd2b78a97c088222d14a.gif . Then http://data.artofproblemsolving.com/images/latex/5/5/3/5536fd12b5f28e7a1f81673b47dd919f3a5f27d6.gif . If http://data.artofproblemsolving.com/images/latex/0/3/f/03f7d5a32f20b3e09718a39837b468e1075fdff8.gif hits http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif at http://data.artofproblemsolving.com/images/latex/6/8/6/686cf6635679037d0da6c7180e2943ac3b9e23e6.gif , we immediately get http://data.artofproblemsolving.com/images/latex/9/8/b/98bc72d077f73e3a1df9623a638ebb300954fbcb.gif is cyclic and http://data.artofproblemsolving.com/images/latex/9/d/4/9d4fc26a50bdeb6214f335053db2509f2e6393c1.gif are collinear,

Similarly, if http://data.artofproblemsolving.com/images/latex/b/1/f/b1fb3bec6fdb22e19a94fe4c6c4481ccba2ee9f0.gif at http://data.artofproblemsolving.com/images/latex/d/1/6/d160e0986aca4714714a16f29ec605af90be704d.gif , then http://data.artofproblemsolving.com/images/latex/0/e/6/0e652b24136b04a4e6b0c9c54f027a0e1ba8c00b.gif are collinear.

Now, http://data.artofproblemsolving.com/images/latex/3/c/7/3c79a7fae6a8e00fd9a4a606df1498166bf94880.gif is cyclic, which gives http://data.artofproblemsolving.com/images/latex/1/4/4/144e2def31a00854c9a12dc7afd3b54d23d7121b.gif . Since http://data.artofproblemsolving.com/images/latex/5/4/8/5489d6b40014c4bd9c71fd09bb9a95945b63d103.gif is also cyclic, http://data.artofproblemsolving.com/images/latex/8/5/2/8524da9c90c7457f90cb374edee1542404724d3e.gif. But ofcourse http://data.artofproblemsolving.com/images/latex/e/1/4/e1455ae6ae0f45d4cf0a96e69a74334a1c6c85ae.gif.
1.(XmL )My proof doesn't require proving the collinearity that ThirdTimeLucky mentioned(even though it's pretty apparent when drawn). It's clear that http://data.artofproblemsolving.com/images/latex/d/b/6/db6cb0096bc200144861ee6073f8e207f9574b1c.gif are concyclic, since http://data.artofproblemsolving.com/images/latex/a/e/3/ae376f9cbd130e6a792483a938f373223d8ecc2c.gif. This means we only have to prove http://data.artofproblemsolving.com/images/latex/c/3/1/c3156e00d3c2588c639e0d3cf6821258b05761c7.gif is concyclic with http://data.artofproblemsolving.com/images/latex/2/6/1/2611def36987afec6083f8f9bc62a9e3c85ab010.gif . Through http://data.artofproblemsolving.com/images/latex/e/0/1/e0184adedf913b076626646d3f52c3b49c39ad6d.gif we construct a line parallel to http://data.artofproblemsolving.com/images/latex/0/f/4/0f4d56d1e20778bf2e1052ecb3219509238fb660.gif and it intersects http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif at http://data.artofproblemsolving.com/images/latex/c/0/3/c032adc1ff629c9b66f22749ad667e6beadf144b.gif , since http://data.artofproblemsolving.com/images/latex/b/0/6/b06c7e0cde491bafedf4ecaa7ae7f85bf550adfa.gifis a rectangle. Since http://data.artofproblemsolving.com/images/latex/c/3/1/c3156e00d3c2588c639e0d3cf6821258b05761c7.gif is concyclic with http://data.artofproblemsolving.com/images/latex/2/6/1/2611def36987afec6083f8f9bc62a9e3c85ab010.gif and we are done.
Suppose http://data.artofproblemsolving.com/images/latex/5/3/f/53f0fb8cdaf6caf8194037215f8ee39e7acbce20.gif be their corresponding multiplicities. Thus, http://data.artofproblemsolving.com/images/latex/b/0/c/b0cf9ee5a6ee6ebd43836db0178e04e394d7b2d7.gif. 
Suppose: 
http://data.artofproblemsolving.com/images/latex/7/9/7/797abc23557866c5e1e47a520c62f0d876f73926.gif  , where http://data.artofproblemsolving.com/images/latex/a/4/2/a42fcb33c52f1a9ca4ac47cbb560f5b1562333b2.gif

http://data.artofproblemsolving.com/images/latex/8/5/5/8555093873a4bd74395c87ade08d9eee8f9b7087.gif  , where http://data.artofproblemsolving.com/images/latex/9/4/2/9426f1dc62a8dafe94a23a71dece69d230d92760.gif

http://data.artofproblemsolving.com/images/latex/3/1/0/3100265ed9f7a342ef296b6455a3804df5ec4929.gif  , where http://data.artofproblemsolving.com/images/latex/3/5/6/356a8bed1732ec1da841c467e4fd3bed50c083da.gif

It can be checked that:

(i) http://data.artofproblemsolving.com/images/latex/8/7/9/879a219b08583c9a5d64a75825fe77d364238187.gif

(ii) http://data.artofproblemsolving.com/images/latex/4/9/8/498f30a3ccfa9acef3e2421c590d156971f2480c.gif

(iii) http://data.artofproblemsolving.com/images/latex/b/1/f/b1fd140519e54f248d7dd9072f4cbc321f187290.gif

(iiii) http://data.artofproblemsolving.com/images/latex/9/0/b/90b08781d1d2d20d28db60dc2aba9d933e3ff0e0.gif

Therefore http://data.artofproblemsolving.com/images/latex/f/7/9/f795885d4cdc4f7af63f69c25a1372d7ab4ac3a2.gif, since http://data.artofproblemsolving.com/images/latex/0/0/8/008db5316aad917adf19796dcf6926c73f6532d7.gif.

Sorry for being a little bit sloppy in the details. I think, I fixed it.
Comment. Let us denote http://data.artofproblemsolving.com/images/latex/a/6/6/a665233bc5e0334806218f28cfe4efd00920bdfe.gif. It is interesting to note that http://data.artofproblemsolving.com/images/latex/4/1/2/4120bff880f393cdb5932e895eb7cc1d1a3c031e.gif and it is a sharp estimate; the equality holds when http://data.artofproblemsolving.com/images/latex/6/d/c/6dcd4ce23d88e2ee9568ba546c007c63d9131c1b.gif is an arithmetic progression. Also, knowing nothing apriori about http://data.artofproblemsolving.com/images/latex/7/c/4/7c42ee326ede4b31ed01e6560fe1f314758e6718.gif, we can only claim http://data.artofproblemsolving.com/images/latex/4/6/4/464f8a3af024eb8311f78b4ddea64ce91008cb3a.gif , since the equality holds when http://data.artofproblemsolving.com/images/latex/0/4/2/042c899ee0670fd4d095cfdad599dfd561b424f1.gif is an arithmetic progression. But nevertheless, as the problem claims, http://data.artofproblemsolving.com/images/latex/a/f/0/af00bf4bcb87a90ab8505b00703afc2f45e98841.gif. Roughly speaking, it is due to impossibility http://data.artofproblemsolving.com/images/latex/6/d/c/6dcd4ce23d88e2ee9568ba546c007c63d9131c1b.gif and \{\ln b \mid b\in B_0 \} simultaneously be arithmetic progressions.
Proof: Note that when http://data.artofproblemsolving.com/images/latex/4/1/9/419c4295a265c6cba14e4237d697ef6a1948ee28.gif, then http://data.artofproblemsolving.com/images/latex/f/3/6/f36c1bc88cc6de0b322bb59979985aed600f3c9d.gif
Construct the following infinite sequence of numbers: http://data.artofproblemsolving.com/images/latex/3/4/6/34663d14eefe0496d8967d0cd14b500a09fed220.gif. Note that http://data.artofproblemsolving.com/images/latex/3/d/5/3d5553eaf7e7b3bcb1a92ebb623de8951fdd32ce.gif for all http://data.artofproblemsolving.com/images/latex/6/f/e/6fe631d550f98de6c2527d4d858f038604fa234d.gif, and http://data.artofproblemsolving.com/images/latex/e/3/c/e3cbe635e202b5e06d6f155608ae24cf4e2271da.gif for all http://data.artofproblemsolving.com/images/latex/3/c/e/3ced74cab564cd52f7d97cced08fe9700afff503.gif for all http://data.artofproblemsolving.com/images/latex/e/e/b/eebecf421c4d33eeab4a0c4da6c20ed8d49e6c6c.gif. Hence there are infinitely many http://data.artofproblemsolving.com/images/latex/2/f/1/2f17c46d99321acaa8de5fcfad1c853e8b55609c.gif. Now for any http://data.artofproblemsolving.com/images/latex/1/1/f/11f6ad8ec52a2984abaafd7c3b516503785c2072.gif, suppose there exist prime http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif such that http://data.artofproblemsolving.com/images/latex/9/a/7/9a75dcf2deb37acd35c82e413bcc496d62f7f613.gif. We can find http://data.artofproblemsolving.com/images/latex/d/0/c/d0ccf51cfe6728032bed8393af726ff5bbfe6537.gif, such that http://data.artofproblemsolving.com/images/latex/6/c/6/6c63e602a0ef08ecaba25812c1f34c9336ed3896.gif, and http://data.artofproblemsolving.com/images/latex/b/5/8/b584458444c4d405a40678abb3e67e208d883052.gif. Then by Chinese Remainder Theorem, there exist http://data.artofproblemsolving.com/images/latex/4/d/c/4dc7c9ec434ed06502767136789763ec11d2c4b7.gif such that http://data.artofproblemsolving.com/images/latex/8/d/f/8df9ba157d146318863524a022d95506dacc5d2c.gif We conclude that http://data.artofproblemsolving.com/images/latex/3/1/6/316936d278b6679d119b35528520e1fb95eaaa64.gif. But http://data.artofproblemsolving.com/images/latex/c/5/c/c5cdc3d3e86752cf831766494880e2c30abca120.gif, contradiction.

Remainder of solution
Note that for http://data.artofproblemsolving.com/images/latex/b/2/0/b208abfdaf0a40f6093c78bf37ac6300ece1d718.gif, suppose there exist http://data.artofproblemsolving.com/images/latex/3/4/4/34498b9401a5a771f51ef614ed7be2b46e50e692.gif. Then http://data.artofproblemsolving.com/images/latex/e/0/f/e0f54ca924dcdbd2fd06c1ab379384a506185fc7.gif, so http://data.artofproblemsolving.com/images/latex/e/f/f/eff683f88d83e21ea85c7937301f960d884207da.gif, contradiction, hence http://data.artofproblemsolving.com/images/latex/3/e/0/3e03f4706048fbc6c5a252a85d066adf107fcc1f.gif is injective for http://data.artofproblemsolving.com/images/latex/7/3/7/737b45b9a8f2b2fea357675c90c769ca2c2f6775.gif. 

Hence if we show that there exist http://data.artofproblemsolving.com/images/latex/a/6/4/a6464f6f31a378bca7899feaf2d699209979434f.gif, such that http://data.artofproblemsolving.com/images/latex/6/3/2/632d96b7f3ab0d65a6f61b523e0a7d0db386bc07.gif for all http://data.artofproblemsolving.com/images/latex/d/0/c/d0ccf51cfe6728032bed8393af726ff5bbfe6537.gif, then http://data.artofproblemsolving.com/images/latex/f/1/d/f1dbdd3a5b38fafd9f78c329171bfe72dea6faf8.gif for all http://data.artofproblemsolving.com/images/latex/7/a/a/7aa0d5db36fd7b9800ad3c90cecb6d9afcc29e45.gif. It remains to show that there are infinitely many such http://data.artofproblemsolving.com/images/latex/3/c/3/3c363836cf4e16666669a25da280a1865c2d2874.gif. Consider http://data.artofproblemsolving.com/images/latex/3/c/3/3c363836cf4e16666669a25da280a1865c2d2874.gif such that http://data.artofproblemsolving.com/images/latex/5/7/0/57005dd7245bdc1e4ddb5941a70fdd09e5605597.gif for all http://data.artofproblemsolving.com/images/latex/4/0/6/406158a3dfa33eda1262bbe11614ca4a289862eb.gif are large enough distinct primes. http://data.artofproblemsolving.com/images/latex/3/c/3/3c363836cf4e16666669a25da280a1865c2d2874.gif exists from Chinese Remainder Theorem. Then http://data.artofproblemsolving.com/images/latex/f/4/a/f4a5baac6440b9492ad09775dcabc1101fcd0505.gif for all http://data.artofproblemsolving.com/images/latex/d/0/c/d0ccf51cfe6728032bed8393af726ff5bbfe6537.gif, unless http://data.artofproblemsolving.com/images/latex/8/e/4/8e4587fc82ce6377530643c5622b41e53cdf3dd3.gif. Hence from lemma we get http://data.artofproblemsolving.com/images/latex/6/3/2/632d96b7f3ab0d65a6f61b523e0a7d0db386bc07.gif for all http://data.artofproblemsolving.com/images/latex/d/0/c/d0ccf51cfe6728032bed8393af726ff5bbfe6537.gif. As we can always pick larger primes, we can have infinitely many such http://data.artofproblemsolving.com/images/latex/3/c/3/3c363836cf4e16666669a25da280a1865c2d2874.gif.
   
注：（USA TSTST 2012, Problem 3）Let http://data.artofproblemsolving.com/images/latex/5/3/6/536c886d7863df5a4e250a73547be5d968c290c7.gif be the set of positive integers. Let http://data.artofproblemsolving.com/images/latex/b/f/e/bfefc53fb39bf125cfb72d6037e1773cbbfe02c2.gif be a function satisfying the following two conditions: 
(a) http://data.artofproblemsolving.com/images/latex/a/5/3/a53ffd447bcf898b6cc05d3a4f5cf05db89f6f08.gif are relatively prime whenever http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif are relatively prime.
(b) http://data.artofproblemsolving.com/images/latex/d/6/a/d6a6cdca46951be252dbe1b87712ef5ee3a035d5.gif for all http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif.
Prove that for any natural number http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif and any prime http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif, if http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif divides f(n) then http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif divides http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif.                          
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）中国数学奥林匹克国家集训队集训2014年3月10日至25日在南京师范大学附中江宁分校举行
                         第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答


              High School Affiliated to Nanjing Normal University ,  12 Mar 2014
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答
第55届（2014）IMO中国国家队选拔考试一（第一天）试题及其解答


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