从1,2,3...,99这999个数中,要求划去尽量少的数,使得余下的数中每一个数都不等于另外两个数的乘积.应划

爱你一生一世

给你公式自己算吧:1+2+3+...+n=n(n+1)/2 n+(n+1)+(n+2)+(n+3)+...+(n+m)=(2n+m)(m+1)/2

牛津英语5A英语语法(四)现在进行时
一、概念
现在进行时表示说话时正在进行或发生的动作,也可表示当前一段时间内的活动或现阶段正在进行的动作．
结构：助动词 be ( am / is / are ) +现在分词.
二、 现在分词的构成：
1.大多数动词后可在动词后直接加-ing．
Eg: carry-carrying,catch-catching,drink-drinking, enjoy-enjoying hurry-hurrying ,do-doing , read-reading , think-thinking
2. 如果动词以-e结尾,则去掉-e,再加-ing,
如come-coming , have-having , make-making,ride-riding,write-writing,take-taking,use-using．
3. 如果动词只有一个元音字母,而其后跟有一个辅音字母时,将此辅音字母双写,再加-ing
如： hit-hitting,let-letting, put-putting,run-running,sit-sitting．
4. 如果动词有两个音节,且重音在第二个音节上,则末尾的辅音字母须双写,再加-ing,
如: for’get-forgetting,pre’fer-preferring,up’set-upsetting．试比较 ’benefit/benfiting, ’differ/differing,’profit/profiting,这些词的重音在第一个音节上,因此其末尾的辅音字母不双写．
5. 以 -ic 结尾的动词,应先把 -ic 变为 -ick,再加 -ing,eg: panic/panicking,picnic/picnicking,但 lie/lying ,die/dying,tie/tying是特殊变化要记住．
三、句型结构：
1.现在进行时的肯定形式、否定形式、疑问形式及其回答,所有变化都体现在助动词 be ( is / am / are ) 上．
1)现在进行时的肯定形式：主语+be(am/ is/are)+doing+其他成分
I am singing . They are writing .
2)现在进行时的否定形式:主语+be(am/ is/are)+not +doing+其他成分
I am not singing . They aren’t writing .
3）一般疑问句及回答：be(am/ is/are)+ 主语+doing+其他成分
Am I singing ? Yes ,you are . / No ,you aren’t .
Are they writing ? Yes ,they are . / No ,they aren’t .
4)特殊疑问句及回答:特殊疑问词+be(am/ is/are)+主语+doing+其他成分
What are you doing ? We are playing (要求就提问内容具体回答).
2. 缩写形式如下:
I am---I’m You are---You’re He is---He’s She is---She’s
It is---It’s We are---We’re They are---They’re
3.说明: 不是所有动词都能用现在进行时态的,如：
see、like、want、know 等动词往往都不用进行时态．
四．用法：
1.表示现在( 指说话人说话时) 正在发生的事情．往往与 now,at the moment,just 等副词连用,以示强调．
We are waiting for you. What are you doing? Some one’s knocking at the door．
2.正在进行着的动作可视为未完成的动作：
He’s talking to his friends in the classroom.
可用 still 一词强调动作的持续性
He’s still talking to his friends in the classroom.
3. 表示长期的或重复性的动作,说话时动作未必正在进行．
Mr. Black is writing another article.
Don’t take that book away. Your father’s using it.
She is learning piano under Mr. Black.
4.现在进行时可用来表示不会长期发生的动作或情况,或被认为在短期内正在进行的动作或存在的情况：
What’s your brother doing these days? He’s studying English at Oxford University.
5.现在进行时也可以用来表示当前的动向：
People are becoming more and more beautiful these days.
6. 表示渐变的动词有：become,turn,get,grow,run,go,begin等．
The leaves are turning brown.
It’s getting colder and colder.
7.与always,constantly,forever 等词连用,表示反复发生的动作或持续存在的状态,往往带有说话人的主观色彩．
You are always changing your mind.
8. 现在进行时(以及 be going to）可以表示为将来安排好的活动和事件 We’re spending next winter in China. 用arrive,come,go,leave 等动词的现在进行时描写行 程安排,也通常有“将到达”和“将离去”的意思： He’s arriving tomorrow morning.
9.当现在进行时表示某事发生的次数过多时,则有时含有抱怨,讨厌,赞扬等的意思：
He is always singing at night,and we can’t fall asleep late at night.

练习
一.用现在进行时完成下列句子:
1. ______you__________(fly) a kite? Yes,_______.
2. ______you___________(sit) in the boat?
3. ______he_____________(talk) with me?
4. We_______________(play) football now.
5. What_________you__________(do)?
6. I_____________(sing) an English song.
7. What________he____________(mend)?
8. He______________(mend) a car.
9. These boys _________ (play) tennis on the playground.
10. My mother______________ (cook) in the kitchen.
11. We can’t help you,because we ____________ (have )classes.
12. ________ the boy ___________ (write) his homework?
13. Look! These butterflies _________ (fly) in the sky.
14. Listen! The girl ___________ (sing) in the next room.
15. The naughty boy __________ (swim) in the river.
二.选择
1. Look. Lucy is_____ a new bike today.
A. jumping B. running C. riding D takeing
2. The children _____ football.
A. is playing B. are playing C. play the D. play a
3. They ______TV in the evening. They do their homework.
A. are watching B. can’t watching C. don’t watch D. don’t watching
4. Listen! She____ in the classroom.
A. is singing B. sing C .to sing D. is sing
5. ______are you eating? I’m eating ______ meat.
A. What,some B. Which,any C. Where,not D. What,a
6. Is she ____ something?
A. eat B. eating C. eatting D. eats
7.My dictionary ___,I have looked for it everywhere but still___ it.
A. has lost,don’t find B. is missing,don’t find C. has lost,haven’t found D. is missing,haven’t found.
8..Having a computer for personal use is no easy task because technology _______ so rapidly.
A. is changing B. has changed C. will have changed D. will change
9. The building_______ ,I can’t stand the noise.
A. was being built B. is built C. is being built D. builds
10. I can’t catch up with the fashion,because the clothes style_______ all the time.
A. has changed B. is changed C. is changing D. changed
11. It’s six in the afternoon. The Greens_______ lunch together.
A. has B. are having C. have had D. had had
12. Don’t make any noise while the students_______ to the class.
A. are listening B. listened C. have listened D. had listened
13. Jack and Ketty_______ in the lake. Let’s join them,shall we?
A. swim B. have swum C. swam D. are swimming
14. Look! The children_______ basketball on the playground.
A. plays B. played C. is playing D. are playing
15. The kite_______ high in the sky now. It looks like a big bird.
A. has flown B. is flying C. was flying D. flew
16.As we all know,the population in the world _______ faster and faster.
A. is grown B. is growing C. are grown D. are growing
17. To my surprise,he_______ in class.
A. is always speaking B. would always speak C. has always been speaking D. does speak always
18. I want to know when he _______ for New York tomorrow.
A. has left B. is leaving C. had left D. has been leaving
19. He_______ of how he can do more for the people.
A. had always thought B. is always thinking C.has always been thought D. thinking always
20. Forests _______ and burned at such a speed that they will disappear from the earth in the future.
A.cut B. are cut C. are being cut D. had been cut
1. Are ,flying,I am 2. Are sitting, 3.Is,talking 4. are plying 5. are,doing 6.am singing 7. is,mending 8. is mending 9.are plying 10.is cooking 11. are having 12. Is ,writing 13. are flying 14.is singing, 15. is swimming
1.C 2.B 3.C 4.A 5.A 6.B 7.D 8.A 9C 10.C 11.B 12.A 13.D 14.D 15.B 16.B 17.A 18.B 19.B 20.C

1层的完全二叉树,有2^0=1个结点（=2^1-1)
2层的完全二叉树,有2^0+2^1=3个结点=(2^2-1)
3层,有2^0+2^1+2^2=7个结点=(2^3-1)
显然,按照上述规律,大于等于999,并且最接近的数是1024(=2^10-1)
答案：深度为10层.

平均数乘每组的个数,就是每组数的和
20组数的和,就是0--999这1000个自然数的和
平均数：
（0+1+2+.+999）÷1000
=(0+999)×1000÷2÷1000
=999÷2
=499.5

A+B=0.00……0(999个0)14
A-B=0.00…0（1000个0）6
A×B=0.00…0（1999个0）4
A÷B=2.5

ab=1*10﹣1995（次方）
a/b=1.25/4=0.3125

10的999+1次方*2-1 第n个数为 10的n次方*2-1

折给你999个小星星,是因为我喜欢你!
Give you 999 stars, is because I love you!

牛津英语5A英语语法(四)现在进行时
一、概念
现在进行时表示说话时正在进行或发生的动作,也可表示当前一段时间内的活动或现阶段正在进行的动作．
结构：助动词 be ( am / is / are ) +现在分词.
二、 现在分词的构成：
1.大多数动词后可在动词后直接加-ing．
Eg: carry-carrying,catch-catching,drink-drinking, enjoy-enjoying hurry-hurrying ,do-doing , read-reading , think-thinking
2. 如果动词以-e结尾,则去掉-e,再加-ing,
如come-coming , have-having , make-making,ride-riding,write-writing,take-taking,use-using．
3. 如果动词只有一个元音字母,而其后跟有一个辅音字母时,将此辅音字母双写,再加-ing
如： hit-hitting,let-letting, put-putting,run-running,sit-sitting．
4. 如果动词有两个音节,且重音在第二个音节上,则末尾的辅音字母须双写,再加-ing,
如: for’get-forgetting,pre’fer-preferring,up’set-upsetting．试比较 ’benefit/benfiting, ’differ/differing,’profit/profiting,这些词的重音在第一个音节上,因此其末尾的辅音字母不双写．
5. 以 -ic 结尾的动词,应先把 -ic 变为 -ick,再加 -ing,eg: panic/panicking,picnic/picnicking,但 lie/lying ,die/dying,tie/tying是特殊变化要记住．
三、句型结构：
1.现在进行时的肯定形式、否定形式、疑问形式及其回答,所有变化都体现在助动词 be ( is / am / are ) 上．
1)现在进行时的肯定形式：主语+be(am/ is/are)+doing+其他成分
I am singing . They are writing .
2)现在进行时的否定形式:主语+be(am/ is/are)+not +doing+其他成分
I am not singing . They aren’t writing .
3）一般疑问句及回答：be(am/ is/are)+ 主语+doing+其他成分
Am I singing ? Yes ,you are . / No ,you aren’t .
Are they writing ? Yes ,they are . / No ,they aren’t .
4)特殊疑问句及回答:特殊疑问词+be(am/ is/are)+主语+doing+其他成分
What are you doing ? We are playing (要求就提问内容具体回答).
2. 缩写形式如下:
I am---I’m You are---You’re He is---He’s She is---She’s
It is---It’s We are---We’re They are---They’re
3.说明: 不是所有动词都能用现在进行时态的,如：
see、like、want、know 等动词往往都不用进行时态．
四．用法：
1.表示现在( 指说话人说话时) 正在发生的事情．往往与 now,at the moment,just 等副词连用,以示强调．
We are waiting for you. What are you doing? Some one’s knocking at the door．
2.正在进行着的动作可视为未完成的动作：
He’s talking to his friends in the classroom.
可用 still 一词强调动作的持续性
He’s still talking to his friends in the classroom.
3. 表示长期的或重复性的动作,说话时动作未必正在进行．
Mr. Black is writing another article.
Don’t take that book away. Your father’s using it.
She is learning piano under Mr. Black.
4.现在进行时可用来表示不会长期发生的动作或情况,或被认为在短期内正在进行的动作或存在的情况：
What’s your brother doing these days? He’s studying English at Oxford University.
5.现在进行时也可以用来表示当前的动向：
People are becoming more and more beautiful these days.
6. 表示渐变的动词有：become,turn,get,grow,run,go,begin等．
The leaves are turning brown.
It’s getting colder and colder.
7.与always,constantly,forever 等词连用,表示反复发生的动作或持续存在的状态,往往带有说话人的主观色彩．
You are always changing your mind.
8. 现在进行时(以及 be going to）可以表示为将来安排好的活动和事件 We’re spending next winter in China. 用arrive,come,go,leave 等动词的现在进行时描写行 程安排,也通常有“将到达”和“将离去”的意思： He’s arriving tomorrow morning.
9.当现在进行时表示某事发生的次数过多时,则有时含有抱怨,讨厌,赞扬等的意思：
He is always singing at night,and we can’t fall asleep late at night.

练习
一.用现在进行时完成下列句子:
1. ______you__________(fly) a kite? Yes,_______.
2. ______you___________(sit) in the boat?
3. ______he_____________(talk) with me?
4. We_______________(play) football now.
5. What_________you__________(do)?
6. I_____________(sing) an English song.
7. What________he____________(mend)?
8. He______________(mend) a car.
9. These boys _________ (play) tennis on the playground.
10. My mother______________ (cook) in the kitchen.
11. We can’t help you,because we ____________ (have )classes.
12. ________ the boy ___________ (write) his homework?
13. Look! These butterflies _________ (fly) in the sky.
14. Listen! The girl ___________ (sing) in the next room.
15. The naughty boy __________ (swim) in the river.
二.选择
1. Look. Lucy is_____ a new bike today.
A. jumping B. running C. riding D takeing
2. The children _____ football.
A. is playing B. are playing C. play the D. play a
3. They ______TV in the evening. They do their homework.
A. are watching B. can’t watching C. don’t watch D. don’t watching
4. Listen! She____ in the classroom.
A. is singing B. sing C .to sing D. is sing
5. ______are you eating? I’m eating ______ meat.
A. What,some B. Which,any C. Where,not D. What,a
6. Is she ____ something?
A. eat B. eating C. eatting D. eats
7.My dictionary ___,I have looked for it everywhere but still___ it.
A. has lost,don’t find B. is missing,don’t find C. has lost,haven’t found D. is missing,haven’t found.
8..Having a computer for personal use is no easy task because technology _______ so rapidly.
A. is changing B. has changed C. will have changed D. will change
9. The building_______ ,I can’t stand the noise.
A. was being built B. is built C. is being built D. builds
10. I can’t catch up with the fashion,because the clothes style_______ all the time.
A. has changed B. is changed C. is changing D. changed
11. It’s six in the afternoon. The Greens_______ lunch together.
A. has B. are having C. have had D. had had
12. Don’t make any noise while the students_______ to the class.
A. are listening B. listened C. have listened D. had listened
13. Jack and Ketty_______ in the lake. Let’s join them,shall we?
A. swim B. have swum C. swam D. are swimming
14. Look! The children_______ basketball on the playground.
A. plays B. played C. is playing D. are playing
15. The kite_______ high in the sky now. It looks like a big bird.
A. has flown B. is flying C. was flying D. flew
16.As we all know,the population in the world _______ faster and faster.
A. is grown B. is growing C. are grown D. are growing
17. To my surprise,he_______ in class.
A. is always speaking B. would always speak C. has always been speaking D. does speak always
18. I want to know when he _______ for New York tomorrow.
A. has left B. is leaving C. had left D. has been leaving
19. He_______ of how he can do more for the people.
A. had always thought B. is always thinking C.has always been thought D. thinking always
20. Forests _______ and burned at such a speed that they will disappear from the earth in the future.
A.cut B. are cut C. are being cut D. had been cut
1. Are ,flying,I am 2. Are sitting, 3.Is,talking 4. are plying 5. are,doing 6.am singing 7. is,mending 8. is mending 9.are plying 10.is cooking 11. are having 12. Is ,writing 13. are flying 14.is singing, 15. is swimming
1.C 2.B 3.C 4.A 5.A 6.B 7.D 8.A 9C 10.C 11.B 12.A 13.D 14.D 15.B 16.B 17.A 18.B 19.B 20.C

解题思路：如果依照题意在上图中进行操作，直到剩下一个数为止，实在是很困难的．我们还应从最简单的情况入手分析，归纳出解决问题的规律，再用此规律解题．
如果是2个数1，2，最后剩下1；如果是3个数1，2，3，最后剩下3；如果是4个数1，2，3，4，最后剩下1；如果是5个数1，2，3，4，5，最后剩下3；如果是6个数1，2，3，4，5，6，最后剩下5；如果是1-7，7个数，最后剩下7；如果是1-8，8个数，最后剩下1．发现当数的个数是2，4，8时，最后剩下的都是1．实际上，当数的个数为2ⁿ时（n≥2），当擦完一圈后还剩2^n-1个数，把问题化成2^n-1个数的情况．不断作下去，最后化为2个数的情况，显然最后剩下的数为1（1为起始数）．

由于2⁹=512，2¹⁰=1024，2⁹＜999＜2¹⁰，
999-512=487．
这就是说，要剩2⁹个数，需要先擦去487个数．按题意，每两个数擦去一个数，当擦第487个数时，最后擦去的数是：487×2=974．
下一个起始数是975，所以，最后剩下的数应是975．
答：最后剩下的数是975．

点评：
本题考点： 哈密尔顿圈与哈密尔顿链．

考点点评： 这类题目有一定的规律：如果数的个数是2n（n是自然数）个数，那么划一圈剩下2n-1个数，划两圈剩下2n-2个数，…划n-1圈，就剩两个数，再划一圈就是剩下安全的数．

1位:9个
2位:90*2=180个
3位:999-9-180=810(个)
共有810/3=270个三位数
所以这本数共
9+90+270=369(页)

研究几个7连乘的积,个位数的规律.
7的个数：1、2、3、4、5、6、7、8……
积得个位：7、9、3、1、7、9、3、1……
显然是7、9、3、1一循环.
999 ÷ 4 = 249 ……余3
因此999个7连乘,积的个位等价于3个7连乘积得个位,为3

两点可以连成一条线段,按照这个思路,
在6个点中选取两个不同的点,一共有C（2,6）=（6*5）÷（2*1）=15种选择,即这么多条线段
同理999个点,选取其中2个不同点即得一个线段,能连成C(2,999)=(999*998)÷（2*1）个线段

31 共30个
一方面 留1,32~999显然可以（32*33>999）；
另一方面,从中分出30组（2,61,122）,（3,60,180）,（4,59,236）,（5,58,290）.（29,34,986）,（30,33,990）,（31,32,992）等30组,显然每组至少要划一个共30个；
故划2~31必为最优解

N为所有正整数的平方.
由大正分成小正,得到的横和竖的个数是一样的,都是正整数个数,乘起来就是平方.
999.1000.1001不行.

1、前999个数的平均数为 (1+999)/2=500
后面1003个数的平均数为(1000+2002)/2=1501
这两个平均数的差是 1501-500=1001

1.1051/2004
1/2004+2003/2004=1.所有分数的和为1001+1/2,中间的分数为1002/2004即1/2,因为剩下的和是502,所以划掉的分数一定包括1002/2004,划掉的分数和为1001+1/2-502=499+1/2,划掉1002/2004左右各499个分数,最大分数为1002/2004+499/2004=1051/2004

123包含在3个数中：1|2|3,1次
123包含在2个数中：1|23：需要开头是23的数,23,231,……239,共10次
或12|3,需要末尾是12的数,12,112,……912,共10次
123包含在1个数中：123,1次
因此最多1+10+10+1=22次

解题思路：由小数的意义：一位小数表示十分之几，计数单位是十分之一；两位小数表示百分之几，计数单位是百分之一，写作0.01，一个小数的计数单位是0.01，它含有999个这样的单位，这个小数是9.99，再加上1个这样的单位它就等于最小的两位数10．

0.01×999=99.9，
10-99.9=0.01，0.01÷0.01=1（个）．
故答案为：9.99，1．

点评：
本题考点： 小数的读写、意义及分类．

考点点评： 此题考查小数的意义和计数单位；应明确一位小数的计数单位是0.1，两位小数的计数单位是0.01．

解题思路：如果依照题意在上图中进行操作，直到剩下一个数为止，实在是很困难的．我们还应从最简单的情况入手分析，归纳出解决问题的规律，再用此规律解题．
如果是2个数1，2，最后剩下1；如果是3个数1，2，3，最后剩下3；如果是4个数1，2，3，4，最后剩下1；如果是5个数1，2，3，4，5，最后剩下3；如果是6个数1，2，3，4，5，6，最后剩下5；如果是1-7，7个数，最后剩下7；如果是1-8，8个数，最后剩下1．发现当数的个数是2，4，8时，最后剩下的都是1．实际上，当数的个数为2ⁿ时（n≥2），当擦完一圈后还剩2^n-1个数，把问题化成2^n-1个数的情况．不断作下去，最后化为2个数的情况，显然最后剩下的数为1（1为起始数）．

由于2⁹=512，2¹⁰=1024，2⁹＜999＜2¹⁰，
999-512=487．
这就是说，要剩2⁹个数，需要先擦去487个数．按题意，每两个数擦去一个数，当擦第487个数时，最后擦去的数是：487×2=974．
下一个起始数是975，所以，最后剩下的数应是975．
答：最后剩下的数是975．

点评：
本题考点： 哈密尔顿圈与哈密尔顿链．

考点点评： 这类题目有一定的规律：如果数的个数是2n（n是自然数）个数，那么划一圈剩下2n-1个数，划两圈剩下2n-2个数，…划n-1圈，就剩两个数，再划一圈就是剩下安全的数．

1,现在将数都看成三位数,不够的补0,例7写成007.
注意到每个数字出现的次数都相等,因为他们是对称的,比如1和2,1能出现的任何地方2也能出现.
所以只需算出来共有多少数字即可.
因为共有1000个数（从000开始）,所以共有3000个数字,除以10,得到每个数字出现300次,所以和就是300*（0+1+2……+9）=13500
2,1000=2^3*5^3
所以与1000互质的数一定没有因子2,5,(因子4,8,10等等也不会有),而没有因子2,则这个数是奇数,没有因子5,则这个数的尾数不能是5,所以这些数应该是 1,3,7,9,11,13,17,19……,991,993,997,999,共400个
求和时为了方便计算,把5,15,……,995,全加上,则组成等差数列,
和是(1+999)*500/2=25万,而5+15+……+995=(5+995)*100/2=5万
所以最后的和应该是25万-5万=20万

（1）
顶点数 边数 区域数
(a) 4 6 3
(b) 8 12 5
(c) 6 9 4
(d) 10 15 6
（2）区域数=边数-顶点数+1
（3）999=边数-999+1
边数=1997

解题思路：如果依照题意在上图中进行操作，直到剩下一个数为止，实在是很困难的．我们还应从最简单的情况入手分析，归纳出解决问题的规律，再用此规律解题．
如果是2个数1，2，最后剩下1；如果是3个数1，2，3，最后剩下3；如果是4个数1，2，3，4，最后剩下1；如果是5个数1，2，3，4，5，最后剩下3；如果是6个数1，2，3，4，5，6，最后剩下5；如果是1-7，7个数，最后剩下7；如果是1-8，8个数，最后剩下1．发现当数的个数是2，4，8时，最后剩下的都是1．实际上，当数的个数为2ⁿ时（n≥2），当擦完一圈后还剩2^n-1个数，把问题化成2^n-1个数的情况．不断作下去，最后化为2个数的情况，显然最后剩下的数为1（1为起始数）．

由于2⁹=512，2¹⁰=1024，2⁹＜999＜2¹⁰，
999-512=487．
这就是说，要剩2⁹个数，需要先擦去487个数．按题意，每两个数擦去一个数，当擦第487个数时，最后擦去的数是：487×2=974．
下一个起始数是975，所以，最后剩下的数应是975．
答：最后剩下的数是975．

点评：
本题考点： 哈密尔顿圈与哈密尔顿链．

考点点评： 这类题目有一定的规律：如果数的个数是2n（n是自然数）个数，那么划一圈剩下2n-1个数，划两圈剩下2n-2个数，…划n-1圈，就剩两个数，再划一圈就是剩下安全的数．

0.000......018*0.000......0111=0.000……01998（中间的省略号表示1999个0）