【文摘】数学难题汇编(4)

1.Show that a graph drawn in the plane such that every face has an even number of edges is bipartite: the vertices can be split into two sets such that each edge connects an element of each set.

1.图论当中,对于一个图,证明:如果每个面都具有偶数条边,则这个图肯定是二分的。二分图指的是可以将点集分成两部分,满足每条边仅存在与两个集合之间。

2.Does there exist a subset of the unit square such that all horizontal lines intersect the set in countably many points but all vertical lines intersect the set in uncountably many points

2.平面当中,是否存在单位正方形的集合,满足所有水平直线与此集合的交点数量是可数的,所有垂直直线与此集合的交点数量是不可数的?

3.One can easily pack 2n circles of unit diameter in a 2xn rectangle. Show that there is some n such that 2n+1 circles may be packed into a 2xn rectangle.

3.可以比较容易的将2n个直径为1的圆不重叠放入2Xn的矩形里面,证明,对于某些n可以将2n+1个直径为1的圆不重叠放入2Xn的矩形里面。

4.Consider the following operation on a rooted tree: Choose a leaf L of the tree with parent P. If P is the root, then truncate L. Otherwise, let G be the parent of P. Duplicate P and the tree below it. Finally, truncate L from both copies of P. Show that for any finite rooted tree, there is a sequence of choices such that this procedure gives the empty tree.

4.对于一个有根的树型,定义下面的操作:
选择一个叶子节点L,它的父节点是P,
i)如果P是根,则剪掉L;
ii)如果P不是根,假设它的父节点是G,则在G下复制增加以P为根的子树,然后将L和复制出的L都剪切掉;
证明,可以选择节点序列,通过以上操作,最后得到空树。

5.Suppose that z is on the unit circle of the complex plane and satisfies a monic equation with integer coefficients. Show that z^n=1 for some n.

5.在复平面当中,如果z位于单位圆周上,并且是一个首系数为1的整系数方程的根,请证明,存在n,使得z^n=1。

6.The circle is a constant-width figure, as is the Rouleaux Triangle (60 degree arcs attached to an equilateral triangle.) Construct one which does not have rotational symmetry.

6.圆形是等宽图形,莱洛三角形也是,能否构造一个非旋转对称的等宽图形呢?

7.Given m>=2 what is the number of subsets of {1,2,3,…,n} that sum to 0 mod m ? 【answers

7.给定整数 m>=2,求集合 {1,2,3,…,n} 的子集数量,满足子集元素的和模m等于0?

8.Prove,that exists a value of n,which number 2^(n) 2 to power n
begins in decimal notation 123454321,
2^(n)=123454321…………
And if exists the value of n then find it.【answers

8.证明:存在整数n,使得 2^(n) 的前面9个数字是 123454321,并具体找到一个这样的n。

9.Will all the 1/k by 1/(k+1) rectangles, for k>0, fit together inside a 1 X 1 square?
Note that the sum of the areas of all these rectangles is 1.

9.是否可以用边长为1/k和1/(k+1)的所有矩形,k>0,铺满1X1的正方形?
注意到:所有这些矩形的面积和等于1.

10.Are there integers n and x (with n>7) such that n!=x^2-1?
By n! we mean the product of the integers from 1 to n. It is known that 4!+1=25=5^2, 5!+1=121=11^2, and 7!+1=5041=71^2.

10.是否存在整数n和x(n>7),满足: n!=x^2-1?
已知: 4!+1=25=5^2, 5!+1=121=11^2, 和 7!+1=5041=71^2 。

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