教学视频-公开课,优质课,展示课,课堂实录(http://www.sp910.com/) Nordic Mathematical Contest 2005 problem 1 Find all positive integers such that the product of the digits of , in decimal notation, equals The product of the digits of is an integer so digits is even; then . Now since is odd, so is odd. is at most (trivial to show) so . On the other hand, it is a non-negative number so because it is odd, that is, . Now we just have to check . Inspection shows both work. , and the product of the is the rightmost digit is even, so Also the product of the digits of problem 2 Let be positive real numbers. Prove that By Cauchy Schwartz By Chebyshev and then Nesbitt, Indeed very easy! I killed this in 10 seconds. Q.E.D. Problem3 There are young people sitting around a large circular table. Of these, at most has a strong position, if, counting from in either direction, are boys. We say that a girl the number of girls is always strictly larger than the number of boys ( is herself included in the count). Prove that there is always a girl in a strong position. It's clear that if the result is true for boys, then it is true for any smaller number 教师之家-免费中小学教学资源下载网(http://www.teacher910.com/) 教学视频-公开课,优质课,展示课,课堂实录(http://www.sp910.com/) of boys (just switch some girls into boys till you have girls and boys. boys, then the result is true and by switching back the boys into girls it stays true). We'll prove the result for This can be done by induction. Suppose the result is true for take girls and boys. Take any girl clear there is one), and then go round the table till you find another have something like the boys in that interval. Among the . . Now take away girls and . Now , so you'll which has a boy for a neighbor (it's where is any of boys left there's a girl in a strong position by hypothesis; it's easy to see she's still strong when we put back Problem4 The circle is inside the circle , and the circles touch each other at . A line through intersects also at , and also at . The tangent to at intersects at and . The tangents of passing thorugh touch at and . Prove that , , and are concyclic. Consider the homotety of center to the tangent of and from here at which tranforms . Hence to then and and we are done. will transform parallel to [相关优质课视频请访问：教学视频网 http://www.sp910.com/] [文章来源：教师之家 http://www.teacher910.com/ 转载请保留出处] 教师之家-免费中小学教学资源下载网(http://www.teacher910.com/) 本文来源：https://www.wddqw.com/doc/89a5111cf211f18583d049649b6648d7c1c70816.html