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    January 24

    难题征解(一)

     
    一,  ABC是一个三条边长都是整数（两两互质）的直角三角形且它的面积也是整数。
    DEF是一个三条边长都是整数的等腰三角形，
    上述两个三角形有相同的周长及面积。

    求ABC及DEF的各条边长（并证明其的唯一性）。
    二,  1）将1---100这100个自然数分成n组，使任何一组中的任何其中两个数之和不是平方数，问n的最小值？
    2）若果将1---m个自然数分成2组，使每组中的其中任何两个数之和不是平方数，问m的最大值？

     

    三,  求证：
    如果一个正整数可以用三个有理数的平方和表示，
    则它也可以用三个整数的平方和来表示。

     

    四,  将1至10000这10000个连续自然数所有的0都擦掉。
    问：余下所有的数的和是多少？
    说明：
    （类似于200、450这样的自然数擦掉0后就变成了2和45，
    而类似于7078这样的自然数则摇身一变成了7和78两个数，10203则变成1，2，3三个数）。


    五,  求证：过三角形各顶点及其重心的三条中线分此三角形成6个小三角形的重心在同一椭圆上。

     

     

    6:35 PM | Blog it | 爱好

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