http://blog.sina.com.cn/s/blog_4c11310201016lup.html

 设正数 http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 满足 http://data.artofproblemsolving.com/images/latex/9/7/7/977056633844dda219742e4eafcf96ad5c8b621c.gif . 求证：

        http://data.artofproblemsolving.com/images/latex/e/1/e/e1e2a5a4253a76e43e0f414fcb7ab5513426dae7.gif

                                        问题421：一个新颖的代数不等式(2013-02-24 09:05:32) 

证明1（arqady，24 Feb 2013, 14:35 ）Let http://data.artofproblemsolving.com/images/latex/5/6/a/56a9a53cac60e119936e5698f2521bddcd6db90d.gif . Hence, http://data.artofproblemsolving.com/images/latex/6/f/0/6f077ff9f11dc9052349f2d9fc7c3fc847773162.gif and

http://data.artofproblemsolving.com/images/latex/b/9/f/b9ff5db169dee58031dbeeb7a21f602d8950b239.gif

http://data.artofproblemsolving.com/images/latex/9/3/c/93c8cdfddaf666d92012f57db18578b84cb06566.gif

http://data.artofproblemsolving.com/images/latex/8/4/2/8427c0371d016f8990f7928728288ffb2f067d03.gif

http://data.artofproblemsolving.com/images/latex/0/2/1/021b272b2f8b8a619f820b40bfb1bd6c2030c838.gif

http://data.artofproblemsolving.com/images/latex/8/3/a/83a8bc981aef6f539e1fc05d10ffd05fea59f73b.gif

http://data.artofproblemsolving.com/images/latex/4/d/2/4d2ecac805ec09df0791ae3dc09633f3badd7bdc.gif

http://data.artofproblemsolving.com/images/latex/d/2/6/d2665950eaac3131a4b1af972bb00afdbdd9fed3.gif.

证明2  （arqady，24 Feb 2013, 14:50 ）The following way is easier.

Let http://data.artofproblemsolving.com/images/latex/c/0/6/c0676799220a8042595a46964f90488dab4e0012.gif , http://data.artofproblemsolving.com/images/latex/9/5/7/957877c9020d275bffa73c7446c3c2d4bb31d190.gif and http://data.artofproblemsolving.com/images/latex/9/a/1/9a1a9692c6f607f0a9e93cb56639458902df221d.gif . Hence, we need to prove that:

http://data.artofproblemsolving.com/images/latex/7/3/b/73bad4a998c4f1da8388c6cdf764a9ebe75a273c.gif, which after full expanding gives:

http://data.artofproblemsolving.com/images/latex/a/b/c/abcc75a6caed7755a38773f8442cf42d57125db4.gif, which is obviously true.

你能改造吗？

    已知 http://data.artofproblemsolving.com/images/latex/8/4/a/84a516841ba77a5b4648de2cd0dfcb30ea46dbb4.gif 是满足 http://data.artofproblemsolving.com/images/latex/e/8/4/e843252a9134677088da0e488ff650bbcc8dbb9c.gif   的实数 . 求证 ：

          http://data.artofproblemsolving.com/images/latex/7/a/5/7a5dd0c80499ab46688dd6ba1d9d28d89d76facf.gif   .
                
                                            问题471:一个三元代数不等式 (2013-03-21 04:25:26)

证明（arqady，21 Mar 2013, 12:56 ）原不等式等价于 http://data.artofproblemsolving.com/images/latex/8/1/5/8150e9d7731eac76069a41f521769c219761aa20.gif .
  

   而 http://data.artofproblemsolving.com/images/latex/b/6/4/b640ef1f9da01fc63bb720991c09a0745dd92e96.gif .

   故只要证 http://data.artofproblemsolving.com/images/latex/6/5/5/655a963d016b8a3122f2b1f66744b6aa44a6ebc4.gif  http://data.artofproblemsolving.com/images/latex/5/b/b/5bb5ddb12eeb1846e1592042f1ac6e9195b7aa49.gif  .

   注  arqady证明的伟大之处可能在于他那有预谋的或下意识的第一步：变形；arqady说：倒数第二式“which is obvious”，最后一步是俺“画蛇添足”的。

  Vasc不等式（ Mar 21, 2013）已知 http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif  是满足 http://data.artofproblemsolving.com/images/latex/3/c/d/3cd5486f66eb8bc7d40be4bd5d0a5a39c1813e4d.gif  的实数,求证：
 

                    http://data.artofproblemsolving.com/images/latex/f/2/1/f21cf7557a3c7d5cf01f159e59ff98df6b487b60.gif