6.    6-Letter Code

You will produce 6-letter codes using the 26 letters of the alphabet.

• A letter can be used only once in a code.
• Any two consecutive letters cannot be used in a single code.

In how many different ways can this be done?

If the same question was asked for the first four letters of the alphabet (A, B, C, D) and 2-letter codes, then the answer would be 6. (AC, CA, AD, DA, BD, DB).

大意就是用26个字母中的不连续的6个字母可以组合成多少种不同的编码组。
他举了一个例子用ABCD4个字母中的不连续的两个可以组成6个组。

7.subject: Arithmetic Mean of the Neighbors

What is the greatest number in which all the digits, except the first and the last digits, are larger than the arithmetic mean of its neighbors? (A neighbor of a digit is the digits to the immediate left and right of that digit)
一个数，除了第一个数码和最后一个数码，中间的数码比前后邻居的均值更大，求这个数。

8.     5-Letter Code

You will use 26 letters of the alphabet and produce 5-letter codes. No vowels will be placed side by side in these codes. How many different codes can be produced?

Note: There are 5 vowels in the alphabet.

Example codes: ABABA, VOCAL, ZZZZZ

用26个英文字母排成长度为5的字符串，但是元音字母不能相邻（总共有5个元音字母），一共有几种不同的排法？

如：ABABA, VOCAL, ZZZZZ

9. Four Bars

You have four bars, which have different integer lengths. You produce a triangle using three of them. When you produce a new triangle, substituting one of the bars with the fourth one, you realize that these two triangles are similar triangles.
What is the possible minimum total length of these four bars?

有4根长度都是整数、长度各不相同的棒，用其中的3根做成一个三角形，而如果用第4根去代替其中的一根，构成的新三角形和原来的三角形相似。求4根棒的长度总和最小是几？

10.  Coins

It is possible to reach every value from 1 to 50 (both inclusive) using at most two of the coins used in a country. What is the possible minimum value of the sum of all coins in this country?

If the same question was asked for values from 1 to 8, the answer would be 8 (1+3+4)

{1}, {1+1}, {3}, {4}, {1+4}, {3+3}, {3+4}, {4+4}

一个国家的硬币系统, 对于任意从1到50的价值, 都可以用至多两个硬币来表示(可重复使用同一个硬币值), 问所有不同价值的硬币的和的最小值是多少?
例子: 如果是从1到8, 和的最小值是8, 可以用如下三个硬币: 1, 3和4.

11. Number Game

You are playing a game with your friend. Your friend chooses three distinct numbers from the set (1, 2, 3, 4, 5, 6, 7, 8, 9). You will call four distinct numbers from the same set in each turn, and he will tell you how many of them are among the chosen numbers.

In order to guarantee to find these numbers in all cases, how many turns are needed?

Note: Naming these three numbers after your 4-number calls are finished does not count as a turn

【数字游戏】
你和朋友玩一个游戏。朋友从1-9中选3个不同的数字(你来猜)。你每轮猜4个不同数字，你的朋友将告诉你猜中了几个他选择的数字。
为了确定他到底是选择的哪3个数字，你需要多少轮？
注：你最后指出他的三个数字时，不算一轮。

12. Ascending Letters

26 letters of the alphabet are shuffled and listed to form a 26 lettered string. The longest sequence of alphabetically ascending letters (either from left to right or from right to left) in this string is noted. What is the minimum possible number of letters in this sequence?

Two examples for the first seven (A, B, C, D, E, F, G) letters:
Arrangement: BGEDFCA, Longest sequence: GEDCA (ascending from right to left).
Arrangement: DBCAGEF, Longest sequence: BCEF (ascending from left to right)

26个字母，
任意不重复排列，
取出其中最长的升序组合（从左往右或者从右往左读均可）
比如 1 2 3 4 5 6 7
随意的一个： BGEDFCA,，其中最长的升序是 GEDCA （从右往左）
随意的一个：DBCAGEF，
其中最长的升序是 BCEF（从左往右）
那么从这26个数字组成的所有排列（26!种，汗。。。）中取出其中最长的升序组合，
最短的一种组合包含多少个数字？

13. Square of the Sums

The square of the sum of four positive integers is equal to the number formed by writing these four numbers side by side. Each of the digits in this number is different. What is the maximum possible value of this number?

四个数之和的平方，等于这四个数紧挨着写在一起组成的新数。这个新数的每一位都不相同，这个新数最大是几？

14. Fifteen Digits

A seven digit number in which the same numeral is used 7 times, divides a 15 digit number in which only numerals 2 and 3 are used. Find this 15 digit number.

一个由单个数码组成的7位数整除一个由数码2和数码3组成的15位数，求这个15位数。

15. Grid

A 2x2 grid can be achieved by drawing three squares on a sheet. How many squares (in minimum) are needed to get a 8x8 grid?

一个2*2的网格，可以通过画3个正方形完成。要完成8*8的网格，最少画几个正方形？

16. Prisms in a Cube

We have a 4x4x4 cube consisting of 64 unit cubes. How many rectangular prisms can be counted in this cube?

Note: Rectangular prisms of every size, including the cubes (and the 4x4x4 cube itself) will be considered.

4*4*4的立方体，包含有64个单位立方体。那么这个大立方体里面有几个长方体？

注：包括每个单位立方体和大立方体本身。

17. Seventeen Intersections

X infinite straight lines are drawn on a plane. There are no three lines intersecting on a single point. If there is a total of 17 intersection points, what can X be at minimum?

If the question was asked for 5 intersection points, then the answer would be 4 (an example drawing is given above).

平面上画了x条直线，任意三条直线不相交于同一点。如果总共有17个交点，那么x最小是多少？

如果是5个交点，则答案是4（上图就是一个例子）。

18.Name and Surname

A robot is given a 4-letter name and 4-letter surname, using the letters of the 26-letter alphabet. None of the letters used in the name is used in the surname. In how many different ways can this be done?

Example: "VGGA NNEN"

给一个机器人取名，姓和名都是分别由4个字母（字母总共有26个）组成。姓中使用的字母不能在名中使用，机器人总共可以取多少种名字？

例如："VGGA NNEN"

19. Regular Tetrahedron

You will select X points within the volume occupied by a regular tetrahedron with edge length of 2 units. The condition is that there should be at least 1 unit distance between each of these points. What is the maximum possible value of X?

Note: The surfaces of the regular tetrahedron (including the edges and the vertices) are considered within the volume.

从棱长为2的正四面体内选择x个点，这些点彼此之间的距离至少为1，问x最大为多少？

注：正四面体的表面（包括棱和顶点）都可以用来取点。

20. A Hundred Students

A hundred students all having different school numbers are arranged as a circle. The difference between the numbers of any pair of students that stand next to each other is at most 10. What is the maximum possible value of the difference between the largest and smallest number?

100个拥有不同学号的同学围成一个圈，相邻同学之间的学号差最大为10，这些同学中最大学号与最小学号之差，最大是多少？

21. Meeting Point

99 athletes (consisting of three groups of 33 athletes) are lined up on a straight line. The first group of athletes are lined up so that one of them will be found in every 103 meters; the second group of athletes are lined up so that one of them will be found in every 107 meters, and the third group of athletes are lined up so that one of them will be found in every 109 meters. (If we set the starting point as 0m, the positions of the first six athletes would be as follows: 103m, 107m, 109m, 206m, 214m, 218m).

After your signal, these 99 athletes will meet on a meeting point you choose. Where should the meeting point be, in order to minimize the total distance taken by all the athletes?

(Enter the distance between the starting point and the meeting point)

99个运动员（每组33人，共三组）排成一条直线，第一组每隔103米排一个，第二组每隔107米排一个，第三组每隔109米排一个，如果我们把起点设在0米处，那么排在最前面的6个运动员分别在103、107、109、206、214、218米处。

你给出一个信号后，这99个运动员会相遇在你选好的集合点上。那么为了使所有运动员走的距离最少，相遇点应该选择在哪里？

22. Painting a Cube

You have six different colors to paint a cube. Each face of the cube should be painted. You can use every color to paint as many faces of the cube as you like (0, 1, 2, 3, 4, 5, 6); but you cannot use more than one color on a single face. How many different painting patterns can be formed on this cube?

Note: If a pattern can be formed by rotating another pattern any number of times, these two patterns are not considered as different.

你用6种颜色不同的颜料去油漆一个正方体，正方体的每一面都要油漆，你可以用任一颜料去油漆任意几个面（0、1、2、3、4、5、6）；但一个面只能用一种颜料。一共有几种方法？

注：如果一种方案可以通过旋转另一种方案得到，则这两种方案是一样的（只能算一种）。

23. Neighborhood Value

Let us define “neighborhood value” as the sum of the products of every consecutive numeral pair with the number of digits between them. (Consecutive numeral pairs are: 0-1, 1-2, 2-3, ..., 8-9).

Example: The neighborhood value for 132 is 2 (1x2x1 + 2x3x0 = 2). There is a single digit between the consecutive numerals 1 and 2, hence 1x2x1. There is no digit between the consecutive numerals 2 and 3, hence 2x3x0. Their sum 2 is the neighborhood value. Likewise, the neighborhood value for 4253 is 2x3x1 + 3x4x2 + 4x5x1 = 50.

Which number with distinct numerals has the largest neighborhood value?

If there exists more than one such number, please enter the largest of these numbers as your answer.

令「邻里值」这个名词的意义为：一个数字当中，其中两个位的数字若为连续数字，则计算该两数之间所夹的数字数量，再求数量与该两数的乘积。然后再寻找下一对连续数字

做同样的运算，直到穷尽为止。而邻里值，即为这些乘积的总和。

例如：132的邻里值为2。

1与2的中间有1个数字→1x2x1=2

2与3的中间有0个数字→2x3x0=0

2+0=2，因此答案为2

例如：试求4253的邻里值为何？

2与3的中间有1个数字→2x3x1=6

3与4的中间有2个数字→3x4x2=24

4与5的中间有1个数字→4x5x1=20

6+24+20=50，因此答案为50

请问：哪一个数字是由不同的数字组成，且具有最大的邻里值？

如果有若干个数字都符合这个要求，请写出其中最大者。

24.Coin Triangles

12 coins are arranged as shown in the figure above. The centers of these coins form various equilateral triangles. (AEF, BJL, DGM, etc.). Your task is to remove minimum number of coins so that no such equilateral triangle remains. Which coins should you remove?

Submit your answer entering the letters of the removed coins in alphabetical order, without any space between the letters.

If you have more than one solution, then enter the solution which comes first alphabetically.

12个硬币排成上图，这些硬币组成了一些正三角形（例如：AEF, BJL, DGM，等等），最少拿走哪几个硬币，使得剩下的硬币不能构成任何正三角形？

如果有几种方法，请给出字母序小的那种。