PREP上的数学题目。。错了很多。谢谢大家帮我解答啦

Q1.S is a finite set of numbers. Does S contain more negative numbers than positive numbers?

(1) The product of all the numbers in S is -1200

(2) There are 6 numbers in S

答案选择E。我觉得应该选择C啊。。

Q2。 70， 75， 80， 85， 90， 105， 130， 130， 130

The list shown consists of the time, in seconds, that it took each of 10 schoolchildren to run a distance of 400 meters. If the standard deviation of 10 running times is 22.4 seconds, rounded to the nearest tenth of a second, how many of the 10 running times are more than 1 standard deviation below the mean of the 10 running times?

答案是2.

可否请大家讲一下解题过程。。

Q3. The function f is defined for all positive integers n by the following rules: f(n) is the number of positive integers each of which is less than n and has no positive factors in common with n other than 1. If p is any prime factor, the f(p)=

答案是p-1

Q4. For any triangle T in the xy-coordinate plane, the center of T is defined to be the point whose X-coordinate is the average (arithmetic mean) of the X-coordinates of the vertices of T and whose Y-coordinates of the vertices of T. If a certain triangle has vertices at the point (0,0) and (6,0) and center at the point(3,2), what are the coordinates of the remaining vertex?

可否请大家讲一下。一直没读懂。

Q5. If 500 is the multiple of 100 that is closet to X and 400 is the multiple of 100 that is closet to Y, which multiple of 100 is closet to x+y?

(1) X<500   (2) Y<400

麻烦请大家讲的时候可否告知一下multiple of X 怎么表达。遇到这类的总是错。。

Q6. Five piece of wood have an average (arithmetic) length of 124 cm and a median length of 140 cm, what is the maximum possible length, in cm of the shortest piece of wood.

这道题的答案给的是100.

我算的只有60。

5*124-4*140这是我的解题过程

Q7. A hiker walked at a constant rate of 4 miles per hour is passed by a cyclist travelling in the same direction along the same path at a constant rate of 20 miles per hour. The cyclist stops to wait for the hiker 5 minutes after passing her, while the hiker continues to walk at her constant rate. How many minutes must be the cyclist wait until the hike catches up?

答案是20min.

谢谢大家。

Q8 At a dinner party, 5 people are to be seated around a circular table. Two seating arrangement are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?

答案是24。这个排列组合的题目总是错。。

第一个题目，举一个反例就发现不能选C比如：

正正正正正负＝1＊2＊3＊4＊5＊（－10）

既符合条件（1）也符合条件2，但是negative numbers没有多于positive numbers

第2个题目，先把所有的时间加起来＝1000然后除以10得出平均数为100

standard deviation是22。4，所以100－22，4＝77，6，小于这个数的都是more than 1 standard deviation below the mean，所以有70和75两个

Q3这个题目首先要明白题目定义的是什么意思，找一个正整数N比N小的正整数和N除了1之外没有公因子，如果P是任何一个质数，先用3算比3小的（1，2，）符合条件那么f（3）＝2，看看下面的选项A：p-1＝2 

B：p－2＝1

C（3＋1）／2＝2

D（3－1）／2＝1

E 2 ，其中有3个答案都可以，说明我们取的这个数字3，不具有普遍性。但是至少可以排除B和D，再看5

同样道理比5小的数字包括（1，2，3，4）。都符合条件，那么F（5）＝4

所以排除了C和E，答案选A

Q6，这个题目你想，中位数是140，平均数是124，要求最短的那个的最大的可能，假设当中3个数字都是140，那么只有最小的和最大的了，而且要保证他们的平均数是124，再想问题求的是最短的那个的最大可能性，所以应该想到数字之间差距越小，那么最小的就可能大一些，最大的可能就是剩下的两段相等，所以用124＊5－140＊3＝200，然后再用200／2＝100，就求出来了。我觉得自己可能说的不太清楚，希望能够帮到你。

Q8这个题目是环形的排列组合，就是比line少1，因为他们首尾相接，所以直接P（4，4）＝24，不用担心，你可以搜索一下，貌似有一个什么linlin的排列组合环形问题总结之类的东西，多做做就好了。加油！！

我感觉Q6 题出的有问题。愚以为

如果 5 块木头长度是 1 ， 140 ，140 ，140， 199  符合中位数是140以及平均数是124的条件。

这种可能也是有的吧？

请高手指正！谢谢

第6题题目问的是最短的那根的最长的可能性的值，所以要使最短的最大，唯一办法是让这几个木头的长度得差别小一点，就是standard deviation 小一点