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几十个十分有用的代数恒等式

(2013-10-15 10:17:24)
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几十个十分有用的代数恒等式
6h404216p3682601
THE-BEAUTY-INSIDE-OF-INEQUALITY几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

https://en.wikipedia.org/wiki/Nested_radicalhttp://tieba.baidu.com/f?ct=335675392&tn=baiduPostBrowser&sc=47836278221&z=2941059694#4783627822155又又http://www.artofproblemsolving.com/Forum/viewtopic.php?f=55&t=404216&55
https://artofproblemsolving.com/community/c6h1208277_use_identities_and_inequalities_in_algebra
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式

http://tieba.baidu.com/f?ct=335675392&tn=baiduPostBrowser&sc=47836278221&z=2941059694#4783627822155又又http://www.artofproblemsolving.com/Forum/viewtopic.php?f=55&t=404216&55


几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

几十个十分有用的代数恒等式

http://tieba.baidu.com/p/1773464093
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式
几十个十分有用的代数恒等式

几十个十分有用的代数恒等式
http://tieba.baidu.com/p/1318934523
几十个十分有用的代数恒等式




2015年全国高中数学联赛讲义之第九讲：常用恒等式与常用不等式
2015年全国高中数学联赛讲义之第九讲：常用恒等式与常用不等式
2015年全国高中数学联赛讲义之第九讲：常用恒等式与常用不等式
2015年全国高中数学联赛讲义之第九讲：常用恒等式与常用不等式

http://data.artofproblemsolving.com/images/latex/1/c/2/1c29dcaaf55c75e849d1e5ebd9b9830ca0dbec44.gif

http://data.artofproblemsolving.com/images/latex/d/6/3/d63b704e270cf21e7e5c8a70c2c8e571512d024f.gif 
If http://data.artofproblemsolving.com/images/latex/3/c/d/3cd5486f66eb8bc7d40be4bd5d0a5a39c1813e4d.gif , then
http://data.artofproblemsolving.com/images/latex/d/c/e/dce09e2760a2dcfe7f1af0c22ee1eb587876a5e9.gif

http://data.artofproblemsolving.com/images/latex/5/c/b/5cb3c67c4c4398c495462f36f6c0c6bfd6ed0003.gif 

http://data.artofproblemsolving.com/images/latex/e/6/e/e6e2cc18cbaefa64357835aa67ef16cf8ea84f3a.gif 
If http://data.artofproblemsolving.com/images/latex/f/e/5/fe50908fc4996fa834d26e97fae39abd04bb27fa.gif ,then
http://data.artofproblemsolving.com/images/latex/2/8/3/283c2e62f10142035e10618d846a478238ba60cd.gif

http://data.artofproblemsolving.com/images/latex/1/f/a/1fae354fc3c3d995b3f6730b0313a27043b06d7c.gif
http://data.artofproblemsolving.com/images/latex/4/2/4/42463d1a7300c03314a1bb42bbc1ab27cf98f4b4.gif
By Ferrari's Identity we have:
http://data.artofproblemsolving.com/images/latex/f/e/8/fe8a5bbd21a715b1f0b63ba39b20b7387c9a1527.gif
If http://data.artofproblemsolving.com/images/latex/a/0/b/a0b3afd6a86cf8fd1bedb3a573c524e68becf251.gif be positive real numbers，thenhttp://data.artofproblemsolving.com/images/latex/5/e/c/5ec905e104ce62780dca3900b4b2a74530488606.gif
http://data.artofproblemsolving.com/images/latex/5/e/d/5ed20940d8a936a6b4d19856cd10bdcd5d6a6e8f.gif .
http://data.artofproblemsolving.com/images/latex/d/e/4/de4b94cb106e3a402873e0994577832414be29d9.gif
http://data.artofproblemsolving.com/images/latex/c/2/e/c2eab060941666bef0f6410aa0b109af1fa15c0f.gif 
where http://data.artofproblemsolving.com/images/latex/3/7/3/373c7648145eca224f2c49bd8b055963cf0ef53e.gif comes from the Sophie Germain identity.
http://data.artofproblemsolving.com/images/latex/5/1/f/51fc2d6187a91a4850bebae781354ec49892ce37.gif
http://data.artofproblemsolving.com/images/latex/1/c/0/1c045779e4d67f57f5ac1f060d34b577645d7403.gif 
http://data.artofproblemsolving.com/images/latex/5/1/6/516329cb1d25e2a82f9c966567c543ddc5268944.gif
http://data.artofproblemsolving.com/images/latex/f/5/9/f59d3cc5db8ab39289d862f1f3cbc0f13f30f4be.gif

http://data.artofproblemsolving.com/images/latex/6/2/4/624dae411270784ae9a2f7178f5c201d405f31fb.gif 

http://data.artofproblemsolving.com/images/latex/0/c/b/0cbf004c6c51cb68aefd06afa009cf561417c5f2.gif 
http://data.artofproblemsolving.com/images/latex/9/f/b/9fbf14db8c2d98295c6627b8e4219bf4aaf78d35.gif 
http://data.artofproblemsolving.com/images/latex/8/e/a/8ea248c516fe335c33f72bd819552b59eb3038c2.gif 

http://data.artofproblemsolving.com/images/latex/5/a/f/5afe43c3b11baef53068c57b6b1763148c698874.gif

http://data.artofproblemsolving.com/images/latex/d/6/7/d6794a789598705b65af5d230493bbf4c4fe639a.gif

http://data.artofproblemsolving.com/images/latex/7/8/d/78d4b7b27215000abd5d6cd5413efeb7ca164122.gif

http://data.artofproblemsolving.com/images/latex/8/0/f/80f462ca723346a142c08ecff719f5acf928f7fc.gif 

http://data.artofproblemsolving.com/images/latex/f/c/4/fc416b69a5092b7952bc2180c48c795881c662ab.gif 

http://data.artofproblemsolving.com/images/latex/5/e/6/5e6d5d1cc020263486f3498712f7c219038a81b7.gif 

http://data.artofproblemsolving.com/images/latex/6/4/9/649bfdfe83ee38de236610e85b13d30d15805320.gif 

http://data.artofproblemsolving.com/images/latex/1/8/3/183135d38730622e7583bbadfae3ac7141928bf3.gif
http://data.artofproblemsolving.com/images/latex/8/c/a/8ca9a2d0a09c6e6965935a190df7797254d34134.gif
http://data.artofproblemsolving.com/images/latex/b/b/f/bbf90f50a43ac102b049695ad1aa342841ee1370.gif
http://data.artofproblemsolving.com/images/latex/5/f/5/5f525c07b095dff83d50a513b47d69ad52590bca.gif
http://data.artofproblemsolving.com/images/latex/b/b/e/bbe7ec4de0f2d3f350c4302751b09525a0136acf.gif
http://data.artofproblemsolving.com/images/latex/b/b/c/bbc6212810ff87c5c07dc0fe350532eb3047e0a5.gif.
http://data.artofproblemsolving.com/images/latex/a/4/6/a46e13ea0caf27154d05e9a3e59ec333e122c5af.gif
http://data.artofproblemsolving.com/images/latex/9/b/0/9b02b7042ed772b76048001008fa765bea57c816.gif
http://data.artofproblemsolving.com/images/latex/5/e/3/5e34ddf44763f46073a4173e20687b43b2a91357.gif
Let http://data.artofproblemsolving.com/images/latex/8/4/a/84a516841ba77a5b4648de2cd0dfcb30ea46dbb4.gif be non-negative real numbers for which http://data.artofproblemsolving.com/images/latex/3/2/9/329a82c37200f943f9ba3e733f7e34605d6d9e7c.gif. Prove that
http://data.artofproblemsolving.com/images/latex/5/b/f/5bf78172ef31052cf4c771fc8b0451ca41ff1a87.gif 
Result 1. - Difference of same powers
http://data.artofproblemsolving.com/images/latex/2/d/b/2dbe6cf7773c8f4b6432de2b05993b7ed3beec8a.gif for non-negative integers http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif and all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 2. - Sum of same odd powers
http://data.artofproblemsolving.com/images/latex/8/3/f/83fd548bb4cfa0937413b7257fe6a28a352c6199.gif for non-negative integers http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif and all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 3. - Special Case of Result 1. - Difference of squares
http://data.artofproblemsolving.com/images/latex/3/4/6/346f4c3f1370cd1d1a6b603d501f4735ba2edf6a.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 4. - Special Case of Result 1. - Difference of cubes
http://data.artofproblemsolving.com/images/latex/4/8/9/48937487f8e9157f8c4e2d805d4ace75662c30d2.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 5. - Sum of Squares - Different representations
http://data.artofproblemsolving.com/images/latex/2/3/b/23bb0344ce4b7998b76d48a1cd61b47bcc5989d4.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 6.
http://data.artofproblemsolving.com/images/latex/f/9/f/f9fae18954ee1c706c77f5f8597e86d80ef3443e.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif .

Result 7.
http://data.artofproblemsolving.com/images/latex/d/7/8/d78bcccd7819acaa8d89b074aa088263ecaa64f2.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif .

Result 8.
http://data.artofproblemsolving.com/images/latex/f/5/2/f52059e9ee4d9a2460a7338658203106ae3f5e0d.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif .

Result 9.
http://data.artofproblemsolving.com/images/latex/5/7/8/5785693e5d81803182b75e0ca69fa89c1364afd8.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif.

Result 10.
http://data.artofproblemsolving.com/images/latex/3/7/9/37968ff15a2fb7bf79fbdafafd7119b3e707d297.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif .

Result 11.
http://data.artofproblemsolving.com/images/latex/a/3/2/a32c1c37aa51b4297f0cfe7c8ac7d83e4beed0d4.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif .

Result 12.
http://data.artofproblemsolving.com/images/latex/7/f/6/7f600dceefc4073ed83616dbb10742088e8e8a6e.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif.

Result 13.
http://data.artofproblemsolving.com/images/latex/5/8/b/58b34b6f9667401d3cdb7cb71b6cbd1200b63860.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif 

Result 14.
http://data.artofproblemsolving.com/images/latex/b/5/b/b5b6fa982f7156a6b6f1f74cf2eb04ae21010301.gif for all reals http://data.artofproblemsolving.com/images/latex/a/9/a/a9a162101845dc0923fb933b81716015f4f16bd2.gif 

Result 15.
http://data.artofproblemsolving.com/images/latex/f/5/2/f52842d9a55394dded2c135fcf9ab5f2710c570a.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 16.
http://data.artofproblemsolving.com/images/latex/8/7/0/87029aa1e2741ae4ca717fa10fd668469a5be7d0.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 17.
http://data.artofproblemsolving.com/images/latex/f/3/2/f3276a7e01f0d1e70e06defb6a240bd18beeb413.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 18.
http://data.artofproblemsolving.com/images/latex/e/1/d/e1dbc6c2a21a87f08f5399a15f0d31b646c9ed3c.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 19.
http://data.artofproblemsolving.com/images/latex/e/8/0/e804934fb665144eceafa436bb45af66b1e0ae07.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 20.
http://data.artofproblemsolving.com/images/latex/1/6/7/167ff3dd4064a0baf5dd5897aeff11e3044a7b73.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 21.
http://data.artofproblemsolving.com/images/latex/e/d/5/ed588874b5833e26c7200003474f6c9d50f1b42c.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 22.http://data.artofproblemsolving.com/images/latex/2/2/1/22193eaad113f412e1568502fb11eae1ef2e29f0.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 

Result 23.
http://data.artofproblemsolving.com/images/latex/4/1/b/41b00f173b5ee7f2d7dc6f6318641feee0e9f345.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 
Result 24.
http://data.artofproblemsolving.com/images/latex/0/3/1/03192615931ce4a03ceb12ef24806902c0cd0ff1.gif for all reals http://data.artofproblemsolving.com/images/latex/9/f/9/9f9ddc81681bf96e846d822b267968e25608095b.gif 
Result 24,5

http://data.artofproblemsolving.com/images/latex/7/d/2/7d2c8eeb615813d72b5ae96d5e2913bd9f9bd87d.gif
Result 25
http://data.artofproblemsolving.com/images/latex/3/6/0/360d0c12daf2820c85991499dcf67cfcda430ded.gif for all real http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 
Result 26
http://data.artofproblemsolving.com/images/latex/9/e/9/9e9a9875353b47d4ad8d0846cfbad6eed0d42db7.gif for all real http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 
Result 27
http://data.artofproblemsolving.com/images/latex/a/f/3/af39df57c5bae4815d01c4fdd93eff60a7fb4ef5.gif for all real http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 
Result 28
http://data.artofproblemsolving.com/images/latex/a/d/7/ad7d7306b8605207ca382db5401be1d1b6457029.gif for all real http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif
Result 30
http://data.artofproblemsolving.com/images/latex/8/0/a/80a660475dc1175cffe3449714d2a600bd888e3c.gif
tbh I don't really know how that could be remotely useful. You might see it once in like every 12387198739136518742687632 problems, who knows? I just saw http://data.artofproblemsolving.com/images/latex/5/3/d/53d1e325bd97e2fa9725c7821fda41b90bd0dc2b.gif and decided to generalize it.
Result31:http://data.artofproblemsolving.com/images/latex/f/1/a/f1a6abff52f6bd99bab88711e35bf3ad76ecd435.gif
Result 33
http://data.artofproblemsolving.com/images/latex/d/6/f/d6f96aebff22305545b9c1bab509f8440d2b26c9.gif
-from Problems from the Book
Result 34
http://data.artofproblemsolving.com/images/latex/9/a/a/9aa0ae7d5b442e18d813d8e0d9758badfdfa767b.gif
-Sophie Germain
What about substitutions? Like:
Result 39: "Lagrange's Identity"

http://data.artofproblemsolving.com/images/latex/e/2/2/e220805e44634a42ce6e47263985433e096cb8fe.gif for all reals http://data.artofproblemsolving.com/images/latex/f/8/1/f81f52413f39694439e5fdc2a543f233f1991052.gif 
Result 40.
http://data.artofproblemsolving.com/images/latex/f/4/c/f4cc2895203fd73ae18206d07c964ffd999c821a.gif
http://data.artofproblemsolving.com/images/latex/b/f/4/bf4df6c6fa2c2f35503cf5968ca916a11ae3319d.gif, for http://data.artofproblemsolving.com/images/latex/8/9/0/890fee35b5ccafd8ef0089032d2e71b53b926496.gif .
Result 41.
http://data.artofproblemsolving.com/images/latex/1/a/a/1aaebfc106aecb98b3e3e65ee28d77c32316542a.gif if and only if one or both of the following conditions are true: http://data.artofproblemsolving.com/images/latex/d/6/7/d6714f1dc0dfc5967c1740d4091cd2113bc03b6f.gif .
Result 42.
http://data.artofproblemsolving.com/images/latex/6/c/4/6c43795bdeadd0dafc90cbcbb11e282ac44b68b2.gif .
Result 43.
If http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif are the sides of a triangle, the identity or inequality can be transformed tohttp://data.artofproblemsolving.com/images/latex/d/1/f/d1ff14b9afef47718969bfc48db4bb6e49c3c313.gif, where http://data.artofproblemsolving.com/images/latex/6/1/d/61d1859c0afe7f5cd628ed510287466a8fb63d23.gif . The converse is also true.

Result 56. http://data.artofproblemsolving.com/images/latex/4/3/5/435370e361199acdcd111947832350b70236b1a1.gif 
  Result 71:
http://data.artofproblemsolving.com/images/latex/3/5/2/352aa90eecff9a024c47ae652472e83527095757.gif
Result 72:
http://data.artofproblemsolving.com/images/latex/7/e/7/7e764fcca71f0de835c77eae1019cc64002f6a17.gif




 

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