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                                       熊昌进 2014西部数学邀请赛第1题的解

参考文献:宋庆2014中国西部数学邀请赛试题及其解答

 

                                  张云华：证2014年中国西部数学邀请赛第1题

 

2.(MrRTI)WLOG arc http://data.artofproblemsolving.com/images/latex/0/f/4/0f4d56d1e20778bf2e1052ecb3219509238fb660.gif be larger than http://data.artofproblemsolving.com/images/latex/b/1/f/b1fb3bec6fdb22e19a94fe4c6c4481ccba2ee9f0.gif so if http://data.artofproblemsolving.com/images/latex/c/3/0/c3096ef2aed111c67aab458e15a1ff8e7d5bc285.gif.

We have http://data.artofproblemsolving.com/images/latex/1/b/9/1b92fd992eb034305fb2356b67a9281b1edeb7a2.gif since they are both isosceles, we get http://data.artofproblemsolving.com/images/latex/f/2/0/f20dd74f1fe14741dea2febd167e1f933a657836.gif. From similar reasoning, we get http://data.artofproblemsolving.com/images/latex/9/5/a/95a3cc361a61fb646d551bc03fab5fc90fc3012c.gif. Hence, http://data.artofproblemsolving.com/images/latex/a/3/c/a3cd5e90f75f48c5152442be853ff004e5bc20d9.gif then multiply to get http://data.artofproblemsolving.com/images/latex/4/0/1/40179b9bd2739c696d0652b154414186192663cc.gif.

But, angle chase gives :
http://data.artofproblemsolving.com/images/latex/f/4/c/f4c5f5d09c49288b965521e8e56d6d1577665ad4.gif
http://data.artofproblemsolving.com/images/latex/5/b/e/5be22ca6b3cff1f3b60dbdf95ce885e83db4e368.gif
Thus, http://data.artofproblemsolving.com/images/latex/0/3/4/034615def4040d3a6c77a230e19f8eb961e61644.gif.
Done.

2.(DVDthe1st)Let http://data.artofproblemsolving.com/images/latex/c/c/4/cc4200da2381358dd8b316a3b2eb7fd765d308af.gif be the midpoints of http://data.artofproblemsolving.com/images/latex/4/9/1/49151900f180c4d187243668254429a2e5925414.gif. Then http://data.artofproblemsolving.com/images/latex/d/d/6/dd6da06b6dcaa2bf8ad087fd0a83cce9026eaee8.gif, hence http://data.artofproblemsolving.com/images/latex/f/a/d/fadae19e6084863f9c88f11483feddc92299d604.gif. Thus http://data.artofproblemsolving.com/images/latex/8/e/0/8e02dce477244bebb6ac842c97522ddb3f0f0b1b.gif and also by anglechasing we can get http://data.artofproblemsolving.com/images/latex/3/e/f/3ef4a0292b09dc61a33e2d9a9b4c24a72a4070a1.gif. Thus we conclude that sincehttp://data.artofproblemsolving.com/images/latex/c/5/3/c535351edb9fbe952ad34c8ad696abdca78effeb.gif, which is equivalent to http://data.artofproblemsolving.com/images/latex/e/4/a/e4a59037cb4b9d6e544f5448c6e3e4a72f42f37e.gif.

2.(BSJL)Suppose http://data.artofproblemsolving.com/images/latex/8/e/8/8e8046e5da9936effaa4b30a032641601b404faf.gif then there are two cases.

Case 1: http://data.artofproblemsolving.com/images/latex/6/a/1/6a12deaf7ee06365f0be783ae7f858463b573bbc.gif     We have http://data.artofproblemsolving.com/images/latex/3/b/d/3bd94e02c556a7a05bbcc2ccbcaca5341656ccf0.gif by using Law of Sines on http://data.artofproblemsolving.com/images/latex/1/7/a/17a07754b2319e690e04b337c6b6c6626b57d25e.gif , respectively.
And now, using Law of Cosines on http://data.artofproblemsolving.com/images/latex/a/a/4/aa4c6106e4f39a785a8b46155a894cb16564d250.gif then we get
http://data.artofproblemsolving.com/images/latex/4/7/4/474df7c38720892ed4a8e87901748b4e9d5cd2d4.gif
Finally, the problem becomes
http://data.artofproblemsolving.com/images/latex/f/b/a/fba02fb630d0012e04ffc61f33a76352a19a0640.gif
 which is not hard!
Case 2: http://data.artofproblemsolving.com/images/latex/8/2/0/820524974670a2a66d1e754cbf3879c052bc4705.gif    It's similar to case 1~((I just too lazy to type

 

                                       平面几何问题1    数学竞赛俱乐部

2014年中国西部数学奥林匹克几何题

张云华：解2014中国西部数学邀请赛试题第4题

2014中国西部数学邀请赛试题及其解答

 6.(MathUniverse)First case: http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif is even. Obviously http://data.artofproblemsolving.com/images/latex/e/e/7/ee7fbeb8da12453abd5243a23a21bb806c8eff9d.gif. If we define http://data.artofproblemsolving.com/images/latex/0/d/1/0d184ccaad74c21421975377d7f96483b1c0ccb6.gif it follows that http://data.artofproblemsolving.com/images/latex/1/5/7/1570c04457762f14da276f6be8b9435954337fcb.gif. Therefore, if n is even, http://data.artofproblemsolving.com/images/latex/a/2/8/a284dd6af59f6aad5b5ac167392401b719ffd1e4.gif.

Second case: http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif is odd, with http://data.artofproblemsolving.com/images/latex/e/a/d/ead5d9b8445b487c6ee0202a02cf465c62daac2e.gif
We will prove that http://data.artofproblemsolving.com/images/latex/1/b/6/1b6e3af117b86cf8fc5e4d0186e1777e0ba01613.gif.
Assume, on the contrary, that there exist numbers http://data.artofproblemsolving.com/images/latex/0/3/2/032b26704771c93972d03ee9770194feea0b5843.gif satisfying condition and http://data.artofproblemsolving.com/images/latex/5/5/d/55d4ab091e6f3d70d6cd64f07a67a121438e9cc3.gif. WLOG, we can assume that http://data.artofproblemsolving.com/images/latex/8/4/a/84acbdd7d60cf0acbf177dd5d80906359be281e5.gif. Then, from assumption we know that http://data.artofproblemsolving.com/images/latex/0/b/3/0b383f67e7e78ff638eda9106465dcea24d9815b.gif. 
Now, assumption implies: http://data.artofproblemsolving.com/images/latex/f/0/8/f083570f3199d581b7b778e4f38c95070bb4ecf8.gif.
Finally, from the above implication, we have:

http://data.artofproblemsolving.com/images/latex/f/0/c/f0c10fccf8500beecd4216aa59ae9ca082fdae17.gif

Contradiction! Therefore, http://data.artofproblemsolving.com/images/latex/3/3/e/33e004c8eef7f756979920950608e0c849a9d63c.gif
However, numbers: http://data.artofproblemsolving.com/images/latex/9/4/b/94b301af74a739f67471855fc18bfeabb543ba91.gif satisfy condition and http://data.artofproblemsolving.com/images/latex/a/8/f/a8fcb1730f3dc2dd5d5452054f1db95577fb4325.gif, so:http://data.artofproblemsolving.com/images/latex/1/b/6/1b6e3af117b86cf8fc5e4d0186e1777e0ba01613.gif. 

               

                                   数学不是算算数，而是启迪心智

    “人生在世，自当狂狷不羁，莫恨穷通修短，西奥数载，且看吾辈！”这不是《古文观止》的名篇，也不是高考满分作文，而是今年中国西部数学邀请赛开幕式上一位女学霸的开幕发言，不仅如此，她还将生涩难懂的文言文发言翻译成英语，让在场外国选手惊叹不已。

　　陈颖洁是重庆育才中学高一(1)班的学生，也是2014年中国西部数学邀请赛育才中学代表队9位选手中唯一的女生。作为发言代表，陈颖洁一上台，先用现代汉语介绍了自己，然后就开始了文言文发言，“山峦嵚崟，雾霭茫茫；两江汇聚，流水汤汤……”

　　接近300字的发言，几乎都是文言文，让现场的选手惊叹不已，发言还未结束，就响起了掌声。不光如此，陈颖洁还在发言结束后，将晦涩难懂的文言文翻译成了英文又说了一遍，让现场外国选手耳目一新。

　　据了解，中国西部数学邀请赛是由中国数学奥林匹克委员会主持的一项数学竞赛。本次大赛汇集了北京、甘肃、广西、新疆等16个省市、香港特别行政区及新加坡、哈萨克斯坦、印度尼西亚、菲律宾等国家共计39个代表队，150多位选手参加。

　　重庆育才中学校长金永说，陈颖洁的发言用了文言文和英语，全面发展，非常好。其实

数学并不止是用来计算的，而是用来启迪心智，让人变得智慧起来的。数学学得好了，心智发展了，各方面都能得到提高，语文、英语等其它学科也不会差。

  重庆育才中学  http://www.cqyc.com/

                               2013中国西部数学邀请赛试题及其解答

 杏坛孔门 2014年中国西部数学奥林匹克邀请赛试题及其解答