【文摘】数学难题汇编(3)

1.Suppose H is a set of real numbers such that
i) 0 and 1 are in H.
ii) For any element a of H other than 0 and 1, there are two elements b and c with average a.
iii) There are finitely many elements of H.
Show that all elements of H must be rational.

1.一个实数集合H,满足
i)包含0和1,
ii)对于任意一个不等于0或1的元素a,存在两个元素b和c,满足a=(b+c)/2,
iii)元素个数只有有限个。
证明:集合H的元素都是有理数

2.Count the number of odd entries in the 100th row of Pascal’s triangle

2.与二项式展开相关的帕斯卡三角形,在第100行当中,有多少个奇数?

3.Does there exist a continuous function on [0,1] which attains each of its values a finite, even number of times?

3.是否存在关于区间[0,1]的映射函数,值域的每个值都只出现有限次,并且都是偶数次?

4.Suppose that each of 3 schools has exactly n students. If each student knows at least n+1 of the 2n students at the other 2 schools, show that there must be a set of 3 students who know each other。

4.有三个学校,每个学校有n个学生,假设每位学生在另外两所学校当中都至少有n+1位朋友互相认识,请证明:肯定存在三位学生,他们互相都认识。

5.What is the greatest product of the parts of a partition of 100

5.将100分解成若干个正数的和,这些分解数的乘积,最大可能是多少?

6.If g(x)=g(x+c) for some x, call c a chord of g. Which real numbers are chords of all continuous functions f with domain [0,1] such that f(0)=f(1)?

6.如果g(x)=g(x+c),对某些x成立,则称c是函数g的一个弦,在区间[0,1]上对于满足f(0)=f(1)的所有的函数f,求哪些数会是它们的弦?

7.Show that if a rectangle can be tiled by a finite number of squares, then the ratio between the lengths of its sides must be rational.

7.如果一个长方形被若干个正方形正好铺满,证明这些正方形的边长比例肯定是有理数。

8.Show that if a rectangle R can be tiled by finitely many rectangles with at least one side rational, then R has at least one side rational.

8.如果一个矩形R能够被有限个矩形铺满,这有限个矩形都至少有一个边长是有理数,证明R也至少有个边长是有理数。

9.Show that if an arithmetic progression (a,a+b,a+2b,…) contains a square, then it contains infinitely many squares.

9.如果在一个算数级数( a,a+b,a+2b,…) 当中,存在一个完全平方数,则一定存在无数个完全平方数。

10.Start with any quadruple of 4 numbers arranged in a cycle. Between each pair of integers write the absolute value of their difference to form the next cycle. Show that upon iteration, any quadruple leads to the all 0 cycle, e.g., (1,2,3,3)->(1,1,0,2)->(0,1,2,1)->(1,1,1,1)->(0,0,0,0).

10.从任何一个循环结构的4元数组开始,取每对相邻的整数计算差值的绝对值,形成一个新的4元数组,请证明,最后都会形成全部是0元素的4元数组。
例如: (1,2,3,3)->(1,1,0,2)->(0,1,2,1)->(1,1,1,1)->(0,0,0,0)

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