【文摘】数学难题汇编(15)

1.任意一个nm+1的序列,必有n+1的增子序列或m+1的减子序列。

2.Let f(m,n) be the maximum possible number of edges in a simple graph on n vertices which contains no m-cycle.
Determine f(m,n)?

2.对于有n个顶点的简单图G,令f(m,n)表示使这个图不包含m-圈的边可能的最大数量,请确定f(m,n)的估值情况。

3.If ab+1, ac+1, and bc+1 are squares?

3.对于三个整数a,b,c,是否ab+1,ac+1和bc+1可以都是完全平方数?

4.No Four Squares In Arithmetic Progression。

4.在等差数列中,是否能够存在连续的4个完全平方数?

5.(Paul Erdos)Divergence implies arithmetic progressions
If the sum of the reciprocals of a set of positive integers is infinite, must the set contain arbitrarily long finite arithmetic progressions?

5.如果一个正整数序列的倒数序列的级数和是发散的,那么这个正整数序列一定包含任意长度的等差数列。

6.Are there only finitely many perfect squares with just two different nonzero decimal digits?
For example, 38^2=1444, 88^2=7744, 109^2=11881, 173^2=29929, 212^2=44944, 235^2=55225, and 3114^2=9696996.

6.是否只存在有限个完全平方数,满足它们仅仅由不同的两个数字组成?
例如: 38^2=1444, 88^2=7744, 109^2=11881, 173^2=29929, 212^2=44944, 235^2=55225, and 3114^2=9696996。

7.Can a closed curve in the plane have more than one equichordal point?
The line joining two points on a curve is called a chord.
A point inside a closed convex curve in the plane is called an equichordal point if all chords through that point have the same length. For example, the center of a circle is an equichordal point for that circle.

7.是否存在一个封闭的曲线,它具有多个等径点。

8.Is there a set S in the plane such that every set congruent to S contains exactly one lattice point?
A lattice point is a point with integer coordinates.

8.在平面上,是否存在一个集合S,满足,任何一个与S全等的集合都正好包含一个格点?

9.Can you find three integers x, y, and z, such that (x+y+z)^3=xyz?

9.是否存在三个整数x,y,z,满足 (x+y+z)^3=xyz?

10.Are there integers n and x (with n>7) such that n!=x^2-1?

10.是否存在整数n和x,其中n>7,满足 n!=x^2-1?

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