测试一

                                             测试二
 
                                            

                                         测试三
                                                      

 转载自  畅想未来

测试一

1. Let the midpoints of sides http://data.artofproblemsolving.com/images/latex/4/2/5/42543a49c4e416a0fb7c5a7626d0d226759c3a22.gif and http://data.artofproblemsolving.com/images/latex/6/2/b/62bd650a8380e4c0ba1db576801b7a1f4725de18.gif be http://data.artofproblemsolving.com/images/latex/c/0/3/c032adc1ff629c9b66f22749ad667e6beadf144b.gif and http://data.artofproblemsolving.com/images/latex/2/3/e/23eb4d3f4155395a74e9d534f97ff4c1908f5aac.gif I will prove that http://data.artofproblemsolving.com/images/latex/2/9/b/29b0da41e95de774bc606b5b2daf7990ce03f740.gif and http://data.artofproblemsolving.com/images/latex/6/9/9/6991e763fbb744296ae44ffd51e0b8e918be3bf4.gif. If this is proved we are done.
First we observe that http://data.artofproblemsolving.com/images/latex/5/a/1/5a1481f74fbd26b1a0dfd808115a3f2f2f39bd33.gif lie on a circle centered at http://data.artofproblemsolving.com/images/latex/b/5/1/b51a60734da64be0e618bacbea2865a8a7dcd669.gif ) and http://data.artofproblemsolving.com/images/latex/9/5/4/954888f64a28c4a4107f3b2737160a4679c9fc9b.gif for the same reason.http://data.artofproblemsolving.com/images/latex/d/7/7/d777ce355a5f4e01bd4269128d6dc5e9dccc132f.gif.
Now let http://data.artofproblemsolving.com/images/latex/f/1/7/f17b49f1aa573ee1fb2b7d35243cb2168a19d4da.gif be the reflection of http://data.artofproblemsolving.com/images/latex/e/6/9/e69f20e9f683920d3fb4329abd951e878b1f9372.gif over http://data.artofproblemsolving.com/images/latex/b/8/4/b84c789e7ba40433347021045ec227ba1cd6a1b0.gif is a parallelogram. Observe that http://data.artofproblemsolving.com/images/latex/6/f/a/6fabf4345e90d09310a9314e40726ba8c27287ce.gif and http://data.artofproblemsolving.com/images/latex/b/9/0/b90a1422efade4ca1cd36c2b333016959b5d8e01.gif
Now we make the important observation that the two figures http://data.artofproblemsolving.com/images/latex/0/9/1/09124391bd0115a06faf17ef125a3982e1823d48.gif and http://data.artofproblemsolving.com/images/latex/9/e/1/9e15be27c655291ddbfe939ef4e9129444d64d3d.gif are homothetic centering http://data.artofproblemsolving.com/images/latex/e/6/9/e69f20e9f683920d3fb4329abd951e878b1f9372.gif mapping http://data.artofproblemsolving.com/images/latex/c/6/3/c63ae6dd4fc9f9dda66970e827d13f7c73fe841c.gif to http://data.artofproblemsolving.com/images/latex/8/8/b/88bc78e0ffa6fea636ade33f84b23ffa5371a8e3.gif to http://data.artofproblemsolving.com/images/latex/a/8/d/a8decd2549ac8a552718779b2158687b47533ced.gif to http://data.artofproblemsolving.com/images/latex/3/2/0/32096c2e0eff33d844ee6d675407ace18289357d.gif and http://data.artofproblemsolving.com/images/latex/b/5/1/b51a60734da64be0e618bacbea2865a8a7dcd669.gif to http://data.artofproblemsolving.com/images/latex/5/1/4/5144d7bcd48f98d157d2a3010e149e31226206b9.gif..Hence http://data.artofproblemsolving.com/images/latex/5/f/f/5ffd9a416bc7a693727cb6a4ff44a8f68934f6bf.gif proving that http://data.artofproblemsolving.com/images/latex/8/2/f/82f3c14eb5e9a25277d593373d82c9b82c4f6c70.gif are concyclic .Done!

2.We will use the following result.
Proposition 1 (corollary of Liouville's theorem) There exists a constant http://data.artofproblemsolving.com/images/latex/c/5/3/c538a6221da718dd38230dcbb6e1a8fb40561f7a.gif such that:

http://data.artofproblemsolving.com/images/latex/f/7/4/f744aace380d7f8e800236140020ab5d3280fdc3.gif  holds for all rationals http://data.artofproblemsolving.com/images/latex/c/b/0/cb0172a656698061aefa8eb30b2dbcd1e31f508e.gif .
Now we are ready to prove the problem. Let us denote http://data.artofproblemsolving.com/images/latex/9/a/7/9a71547564ddc7af038b8016bff9669960c8a53c.gif .
Claim: There exists http://data.artofproblemsolving.com/images/latex/f/a/f/faf7ac9b41aa9eed5beb76067ede74925bf70837.gif such that the count of the numbers http://data.artofproblemsolving.com/images/latex/5/f/e/5fe494e9df3aab28689d3485a14f8141781f20d1.gif between every pair http://data.artofproblemsolving.com/images/latex/e/a/e/eae2d2f3b2de99ad9c8cf313158ba72112fea224.gif is no more than http://data.artofproblemsolving.com/images/latex/4/5/5/455d8ba2837cbe440fcb0dea321c04f6c46cc464.gif .
Proof: Assume on the contrary that for every http://data.artofproblemsolving.com/images/latex/f/8/3/f831eeb2c599d29443ace93b3ee58ea730e6b10e.gif there exists a pair http://data.artofproblemsolving.com/images/latex/e/a/e/eae2d2f3b2de99ad9c8cf313158ba72112fea224.gif s.t. http://data.artofproblemsolving.com/images/latex/f/6/b/f6be31905a96e7ff5e3562bb9b9e0549fa257754.gif
Therefore:
http://data.artofproblemsolving.com/images/latex/8/a/c/8ac3676799e1a8053ca6a71e0965c9761ab5de25.gif
Then pigeonhole yields:
http://data.artofproblemsolving.com/images/latex/3/6/d/36d87de92673c4058330dd7a18bf0358a5096f0d.gif , for some http://data.artofproblemsolving.com/images/latex/2/e/2/2e2defbdaa88384a71adbad31483fa257c433115.gif.
This implies:
(1) http://data.artofproblemsolving.com/images/latex/d/5/9/d597b0fe7d61130209abe154351261659b0439b9.gif

where http://data.artofproblemsolving.com/images/latex/5/7/e/57e468271d2dba0dc973cae788e8e0866d9a62f3.gif

In (1) we can raise http://data.artofproblemsolving.com/images/latex/b/5/1/b51a60734da64be0e618bacbea2865a8a7dcd669.gif as big as we want, which contradics Proposition 1. http://data.artofproblemsolving.com/images/latex/4/a/4/4a4e9e431da45a27bc880a8a1ca44d8b1b9bc143.gif

Now we can set http://data.artofproblemsolving.com/images/latex/b/6/8/b68eee9f53f02b045000fb24aa2c524b7970b4ae.gif .

4.解（shaoxinyu ）  I think the maximum is http://data.artofproblemsolving.com/images/latex/3/5/6/356a192b7913b04c54574d18c28d46e6395428ab.gif
http://data.artofproblemsolving.com/images/latex/7/6/a/76ae90e8b1285284b49533065fd86803604b8533.gif
note that
http://data.artofproblemsolving.com/images/latex/7/f/1/7f1395adc5850849c87b05a88dfaf9ced973cdc2.gif
then we are done
it is obvious that equality holds when

http://data.artofproblemsolving.com/images/latex/b/9/9/b9968652fbb6762df6f671f7fee56bb528907794.gif

测试二

4.Note that for arbitrary integers http://data.artofproblemsolving.com/images/latex/f/f/6/ff60f774c8ad113d35a213ab18e5bc4689a94bed.gif we have http://data.artofproblemsolving.com/images/latex/e/0/a/e0abb2d2f8ddf786ae301d94798c2aa74597fc19.gif is even due to the fact that http://data.artofproblemsolving.com/images/latex/f/1/b/f1b8911934e6ba498ce354aa48962feb341909b3.gif . Now, remark that: http://data.artofproblemsolving.com/images/latex/f/e/3/fe3e16937f46701a43be9b18d2a20a6788365930.gif
However, http://data.artofproblemsolving.com/images/latex/4/9/5/4959627bdb882062bf2d1a1cde4bb119077fd40e.gif of this is congruent to http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif modulo http://data.artofproblemsolving.com/images/latex/d/a/4/da4b9237bacccdf19c0760cab7aec4a8359010b0.gif . It immediately follows that http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif is even.
To motivate this solution, it is clear that to solve this problem we should multiply http://data.artofproblemsolving.com/images/latex/d/f/9/df96d45bf8b68c6eb27a5a94a84281ca39288457.gif together for various http://data.artofproblemsolving.com/images/latex/3/4/e/34e03e6559b14df9fe5a97bbd2ed10109dfebbd3.gif . We want to get a lot of squares of expressions. Thus plugging in some sort of http://data.artofproblemsolving.com/images/latex/7/e/c/7ecc7efb629bb7c598cff65a9d0da7ad990aee9d.gif into the expressions is convenient due to the high level of symmetry present. This gets some http://data.artofproblemsolving.com/images/latex/3/3/7/3370e44ba32a89252d2e400500981d12ac9fe629.gif terms, so http://data.artofproblemsolving.com/images/latex/c/4/c/c4c4cb2a0225653995ee111725cf1a9f0fe1077e.gif would allow us to pull out http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif powers of http://data.artofproblemsolving.com/images/latex/d/a/4/da4b9237bacccdf19c0760cab7aec4a8359010b0.gif 's which is extremely useful. Then plug in http://data.artofproblemsolving.com/images/latex/f/e/2/fe296741abc9ae23f57e3581ea6dfd6a1a52f59d.gif to get only a http://data.artofproblemsolving.com/images/latex/b/8/4/b847b0159e0a830ed76d2de34e787957e3db1b97.gif remaining. Letting http://data.artofproblemsolving.com/images/latex/e/5/e/e5eb05bb23136deb5962f9004ae449e74f7e0547.gif yields the above solution.

6.

                                        2013年集训第四次一道非常好的数列不等式解答

测试三

4.Oh oops, I messed up. I apologize http://www.artofproblemsolving.com/Forum/images/smilies/redface_anim.gif My solution actually works when http://data.artofproblemsolving.com/images/latex/7/9/9/7994e21fd5f2d5a5b20970a80c6689904dfd9b73.gif , here it is.
Let http://data.artofproblemsolving.com/images/latex/c/8/0/c80255fe7589f644b92cb18aea398c5040a5c9d5.gif be two distinct integers in http://data.artofproblemsolving.com/images/latex/8/f/8/8f8e008d6b1c8d81c5287806177e0eab571caf44.gif . I claim that http://data.artofproblemsolving.com/images/latex/5/a/1/5a1db85bcfc931c11de837319313ad40225b9e38.gif. Note this would imply the problem.
To prove this we first prove that if http://data.artofproblemsolving.com/images/latex/9/d/b/9dbcedd0501b152f3bdfb4cb2b9872724c88e8c7.gif then http://data.artofproblemsolving.com/images/latex/2/c/1/2c15fa0baac8923861180f1bb1c4998d39a462e2.gif when http://data.artofproblemsolving.com/images/latex/6/d/c/6dc02c6bbf67284c7b53ff7c8afce433f21d7b7d.gif and http://data.artofproblemsolving.com/images/latex/9/a/6/9a680c834a760ab4d3346647b41ab6a6035d68f4.gif .
Note that by Vandermonde Convolution: http://data.artofproblemsolving.com/images/latex/9/9/5/995a90b2d0993829265fb5d0ab7b5d62fa0bbeaa.gif by the fact that http://data.artofproblemsolving.com/images/latex/9/2/5/925cde0818cd6baeb4c7e2a6e5f5cc6107cff4cb.gif divides http://data.artofproblemsolving.com/images/latex/5/9/1/59190ce66aa154e69843e27af8f52c7a9a952e56.gif . Note that http://data.artofproblemsolving.com/images/latex/2/2/2/2229e2e0cdb7c4fb448f748220c398c0bdd2d9fe.gif , so it follows that http://data.artofproblemsolving.com/images/latex/8/8/2/8828f73196cd96413f5140d13f3a8cd8784fdb36.gif, so it is a unit modulo http://data.artofproblemsolving.com/images/latex/3/9/1/391ae62232b9c5d2991b796daa93f51a289da65b.gif so we can cancel it out.
Thus we require that
http://data.artofproblemsolving.com/images/latex/8/d/2/8d254422dab80ab8c1ba0f5f6a7b1926c86147ae.gif as desired.
Now, suppose that http://data.artofproblemsolving.com/images/latex/9/a/f/9afda08ecea11374ccfa78d2bc5a4766a9aa2d7b.gif. By the lemma letting http://data.artofproblemsolving.com/images/latex/e/3/b/e3b36ada6b3dc57e8ddf294f680bd586bc5f3659.gif we have http://data.artofproblemsolving.com/images/latex/2/c/1/2c15fa0baac8923861180f1bb1c4998d39a462e2.gif so let http://data.artofproblemsolving.com/images/latex/7/c/a/7cafff5f2f0c6ceb59da38f27ad74d925d0e40f3.gif and http://data.artofproblemsolving.com/images/latex/2/d/4/2d40c34f3c9b1a935aac96bdc4d37e182544778d.gif. By the lemma again we have http://data.artofproblemsolving.com/images/latex/b/8/f/b8fb0712b272514818b303d5ebbe512d22e6e2a1.gif so in fact http://data.artofproblemsolving.com/images/latex/2/c/b/2cb9f0eeaf016f098bacf8d0e5e2392626abe19c.gif . By continuing this process and defining the http://data.artofproblemsolving.com/images/latex/3/2/e/32ef2db0ee86abbcc5288478f04863bfb8450c43.gif similarly we yield that in fact http://data.artofproblemsolving.com/images/latex/7/a/9/7a9a1daffd53688953733e9c7f321cd379c5f61b.gif. But this is a contradiction if http://data.artofproblemsolving.com/images/latex/0/7/0/0703f9b3348f23d66bbaf53e15b9da7062fc7713.gif are distinct so we are done. Hence the problem immediately follows, as the numbers of the form http://data.artofproblemsolving.com/images/latex/c/a/4/ca4de6d50172a3a8125eb55356e9de2dd7b2f1a1.gif for http://data.artofproblemsolving.com/images/latex/7/9/3/793e359415f45a398310517dd449f03cbd483a54.gif must cover all residues modulo http://data.artofproblemsolving.com/images/latex/a/1/9/a19fc7d8aab639d6819092fe3bb99d8eae2c8392.gif .

5.

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