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奥数练习题 1:
- 你会用圆规和直尺作出一给定正方形面积的7/8的正方形吗？或面积为2的开放的正方形吗 ？
- Do you know how to use a compass and a ruler to make a square that is 7/8 of the area of a given square or with √2 as area ?
奥数练习题 2：
定义：如果一组四个不同的正整数（a,b,c,d）符合 a*b-c*d=1 这个关系的话，被称为“有趣数集”。
命题：假如有 {1,2,3,4,5,6,7,8… 4*n} 这样一个数集，证明不可能有n组用尽所有4*n个数的“有趣数集” 。
Definition: If a set of four natural numbers (a, b, c, d) satisfies the relationship a*b-c*d=1, it is called an "interesting number set".
Proposition: If there is a set of numbers {1,2,3,4,5,6,7,8… 4*n}, prove that there cannot be n sets of "interesting number sets" that use up all 4*n numbers.
奥数练习题 3：
定义：Sn = (2+3+…n)/1 + (3+4+…n)/(1+2) + (4+5+…n)/(1+2+3)+… + n/(1+2+… (n-1)) , n>1
请证明 Sn 是偶数。
Definition : Sn = (2+3+…n)/1 + (3+4+…n)/(1+2) + (4+5+…n)/(1+2+3)+… + n /(1+2+… (n-1)) , n>1
Prove that Sn is an even number.