The More,the Cheaper

Deciding to do som e back-yard landscaping,m y father-in-law wentto the brickyard to buy a quantity ofbricks.W hen he asked the salesclerkabout the cost,the m an replied,“The m ore you buy,the cheaper theyare.”“Is that so,”said m y fahter-in-law.“Then just keep loading them onm y truck untilthey are free.”The More,the Cheaper…     

An Unforgettable Mid-autumn Day

That was a traditional (传 统的) Chinesefestival—M id-autum n Day. As Senior 3students, we had classes that day. Som efam ilies were sitting around the tables, eatingdelicious m ooncakes. Som e were wandering inthe m oonlight, talking about pleasant things.Daughters and sons were on their way backhom e. But I was away from m y parents,unable to get back hom e. At that verym om ent,I m issed them very m uch,blessing(祈祷)them in m y heart.During the daytim e, I saw a girl walkinghand-in-hand with her father and m...     

An Unforgettable Mid-autumn Day

That was a traditional (传 统的) Chinesefestival—M id-autum n Day. As Senior 3students, we had classes that day. Som efam ilies were sitting around the tables, eatingdelicious m ooncakes. Som e were wandering inthe m oonlight, talking about pleasant things.Daughters and sons were on their way backhom e. But I was away from m y parents,unable to get back hom e. At that verym om ent,I m issed them very m uch,blessing(祈祷)them in m y heart.During the daytim e, I saw a girl walkinghand-in-ha…     

美国《数学杂志》问题1714的一个推广

美国《数学杂志》2005年二月问题征解1714[1]为:设m,n,x,y,z∈R ,且x y z=1.证明:44()()()()x ymx ny my nx my nz mz ny421()()3()z mz nx mx nz≥m n.(1)本文给出了(1)式的一个推广:定理设λ,ai∈R (i=1,2,L n),且a1 a2 L an=1,an 1=a1.则当k≥4或k≤0时,有321(1)(1)(1)nk ki     

狮身人面像

Characters:Blue M ouse,Superboy,R ed M agician,Skeleton Soldier and Sphinx.Scene:in frontofthe Sphinx注释:riddle['ridl]:n.谜语Sphinx[sfi耷ks]:n.斯芬克斯(狮身人面像)character['k覸rikt藜]:n.人物m ouse[m a尬s]:n.老鼠Superboy['sju蘼p藜b蘅i]:n.超级男孩m agician[m藜'd廾i蘩藜n]:n.法师Skeleton Soldier:n.骷髅士兵scene[si蘼n]:n.场景in frontof:在……面前com e outof:走出……G reatPyram id:大金字塔w rong[r蘅耷]:adj.错的,出错的half[h藁蘼f]:adj.一半的,半个的(Blue M ouse,Superboy,Red M agician,and Ske…     

HAPPY English

In a bookshop在书店D ictionary:file 文件夹com ic 漫画new spaper 报纸 postcard 明信片W ord s box: 指示代词 ·this 这(单数) ·that 那(单数) ·these 这些(复数) ·those 那些(复数) Play gam e: 读句子,将图片序号填入括号中,并给 图片涂上相应的颜色。 ( )Look.This is m y black nose. ( )Look at m e.This is m y green ear. ( )Look.This is m y brow n body. ( )This is a purple ruler. ( )This is a pink sharpener. ( )This is an orange pencil. notebook 笔记本dictionary 词典 book 书本 sticker 黏纸 1 2 3 4 5 …     

The Coolest Dad in the Universe

H e was 50years old w henI was born,and a “M r M om ”long before anyone had a nam efor it. I didn’t know why hewas hom e instead of M om ,butI w as young and the only oneof m y friends w ho had theirdad around. I considered (认为)m yself very lucky. D ad did so m any thingsfor m e during m y grade schoolyears.H e convinced(使 ……信服; 使……确 信) the schoolbus driver to pick m e up atm y house instead of the usualbus stop that w as six blocksaway.H e alw ays had m y lunchready for m e…     

小山上的火

1. Long, long ago there was a lake of cold water in Kenya. People never came to the lake at night, for fear that the animals near it should kill and eat them. 久以前,在肯尼亚有个冷水湖。人们从不 在夜里到湖边去,因为他们害怕湖附近 的动物会咬死并吃掉他们。2.N ow ,a rich m an had a beautiful daughter.O nce he said,“The young m an who will stay all night in the lake will have m y daughter for his w ife.” There lived a poor young m an w ho loved the rich m an s daughter very m uch. H e w ould t…     

数学奥林匹克高中训练题(69)

第一试一、选择题 (每小题 6分 ,共 36分 )1 .如果x、y >0 ,logxy logyx =1 03,xy =1 4 4,那么 ,x y的值是 (　　 ) .(A) 2 0 3　　　　 (B) 2 6 3(C) 2 4 3(D) 1 0 32 .设z=m ni是方程az4 ibz3 cz2 idz e=0的一个复数根 ,这里a ,b ,c ,d ,e,m ,n∈R .则下列各数一定是方程的根的是 (　　 ) .(A)m -ni　　　 (B) -m ni(C)n mi　　　 (D) 1 -m ni3.已知等差数列 {an}和等比数列 {bn}的各项都是正数 ,且a1=b1,a2n 1=b2n 1.那么 ,一定有 (　　 ) .(A)an 1 1　　 (D)an 1≥bn 14.在…     

利用点列共线证明等差数列的性质

等差数列的通项可以表示为an=dn+(a1-d),从函数的观点看,点列(n,an)在直线y=kx+b(k=d,b=a1-d)上.故有下面的命题: 命题 若[an]是等差数列,则点列(n,an)在同一条直线上. 设Sn是等差数列的前n项和,易证{Sn/n}为等差数列.由命题知下面的推论成立.     

An Unforgettable Thing

The story happened when I w as ten years old.O n a rainy day,m y father w as busy and he asked m e to go to a shopto buy som ething for him . The shop was about three m iles aw ay from ourhom e.I set out with an um brella in m y hand.The road was very wet andm uddy,and I walked with great difficulty.A fter I bought w hat m y fatherneeded,I started to go back with all the things in m y hand.O n m y w ayhom e I fellseveral tim es,but I w asn tbadly hurt.W hen I was 500m etresaway from our ho…     

也谈一道数学奥林匹克题的简证及推广

1999年加拿大数学奥林匹克竞赛有一道试题 :令 x,y,z是满足 x y z=1的非负实数 .证明 :x2 y y2 z z2 x≤ 42 7,并指出等号成立的条件 .文 [1]给出了这道赛题的简证并将其推广为 :令 x,y,z是满足 x y z=1的非负实数 ,则xnym ynzm znxm≤ nnmm(n m) n m(n>m,n,m∈ N) .上述推广是正确的 ,但赛题和推广的证明方法都是错误的 .这是因为式子xnym ynzm znxm (n>m,n,m∈N) (*)是关于 x,y,z的轮换对称式 ,而不是 x,y,z的 (可换 )对称式 .如果在 (*)式中作轮换代换 (x,y,z)→ (y,z,x)或 (x,y,z)→ (z,x,y) ,所得式子与 (*)式相同 ;但…     

美国《数学杂志》问题1714的再推广

美国《数学杂志》2005年二月问题征解1714:设m,n,x,y,z∈R+,且x+y+z=1,证明:44()()()()x ymx+ny my+nx+my+nz mz+ny421()()3()z+mz+nx mx+nz≥m+n.(1)文[1]将其推广为:设λ,ai∈R+(i=1,2,n),且1niia=∑=1,an+1=a1,则当k≥4或k≤0时,有321(1)(1)(1)nk kii i i i ia naλa aλaλ?=++∑++≥+.本文在文[1]的基础上对(1)式进行再推广:命题1设m,n,x,y,z∈R+,且x+y+z=1,α,β,γ∈R+,且α?(β+γ)=2,则()()()()x ymx ny my nx my nz mz nyαα+β+γ++β+γ1()()3()zmz nx mx nz m nα++β+γ≥+β+γ.命题2设m,n,x,y,z∈R+,且x+y+z=1,β,γ,l∈…     

还原生活 寓学于乐

[案例1]这是译林版《牛津小学英语》3A教材中的一课,内容是用下列句子作简单的自我介绍:M y nam e is….I'm an E ng-lish/a Chinese boy/girl.I'mfrom…I'm nine/…M y hair islong/short.M y eyes are big/sm all.I'm thin/fat.I'm tall/nottall.在巩固阶段,有位老师是这样进行操练的:T:H ello.M y nam g is×××.W hat's your nam e?(对着一位学生)S:M y nam e is×××.T:I'm a Chinese m an.S:I'm a Chinese girl.T:I'm from×××.S:I'm from×××.T:I'm tall.(同时把手举过头顶)S:I'm nottall.(也很机灵地把手举过头…     

为他做同样的事

O ne day, a poor farm er took a bag of w heat to tow n. H e didn’tknow w hatto do w hen the bag felloffhis horse. The bag w as too heavyfor him to lift.H e hoped thatsom eone w ould com e and help him . B efore long a m an cam e over.Butthe farm er’s heartsank w hen hesaw the m an,w ho w as the great m an in tow n.The farm er w as afraid toask for help. A s soon as he cam e up to the farm er, the great m an got off hishorse and said,“Y ou need m y help,don’tyou? L etm e help you. ”…     

一个不等式的推广

文 [1 ]提出一个猜想不等式 :设 x,y,z∈ ( 0 , ∞ ) ,则有xx y yy z zz x≤ 322 . ( 1 )文 [2 ]应用导数给出了证明 ,文 [3]又给出其下界估计xx y yy z zz x>1 . ( 2 )现将其推广 :设 x,y,z∈ ( 0 , ∞ ) ,n≥2 ,则有1 xx y,yy z>yy z,n zz x>zz x,所以n xx y n yy z n zz x>xx y yy z zz x>xx y z yy z x zz x y=1 .再证右端 .当 n=2时 ,由 ( 1 )知 ,不等式 ( 3)显然成立 .现设 n>2 ,…     

这就是我

M y C hinese nam e is Li Zuo,w orks in a bank,and m y m other and m y English nam e is Frank.w orks in C ER IS.M y father is an I m a boy.I m ten years old.I officer,and m y m other is an T his Is M e’’live in X uan W u D istrict.T here engineer.Ihave a…     

新目标英语八年级(下)Review of units1—5单元要点检测题(英文)

(满分:120分)Ⅰ.听力(20分)A.听对话,选出与图画内容相符的选项。(共5小题,每小题1分)B.听句子,选择正确的答语。(共5小题,每小题1分)6.A.A bout an hour.B.Once a week.C.In a week.D.Every night.7.A.Yes,I could.B.Sure.C.You can.D.You have to think.8.A.Not at all.B.I’m very well.C.Thanks.D.Yes,please.9.A.Som etim e.B.Som etim es.C.Last night.D.At lunch tim e.10.A.For five years.B.W ith m y friends.C.Five years ago.D.Five years later.C.听对话,选出能回答该问题的最佳答案。(共5小题,每小题1分)11…     

圆锥曲线两个性质的合并推广

文[1]给出圆锥曲线的一个有趣的性质:在圆锥曲线(等轴双曲线除外)的焦点所在的对称轴上必存在一定点,过该定点的弦被定点分成长为m,n的两部分,使得m12+n12为定值.实际上椭圆、抛物线还有一个有趣的性质:过焦点的弦被焦点分成长为m,n的两部分,则1m+1n为定值.文[2]探讨了上述第一个性质及推广,本文从另一个角度探索其本质.我们注意到焦点也在对称轴上,那么过对称轴上其它点的弦,究意有什么性质呢?笔者通过对该问题的探究,把圆锥曲线的这两个性质合并推广到一般情况.命题1设过定点P(q,0)(q≠0)的直线交抛物线y2=2px(p>0)于两点P1,P2,记│PP1│=m,│PP2│=n,则m12+n12+│2q(│p-mq·)n为定值.证明设P1P2方程为x=ty+q,代入y2=2px得y2-2pty-2pq=0,设P1(x1,y1),P2(x2,y2),则y1+y2=2pt,y1y2=-2pq.而m2=│EP1│2=(1+t2)y12,n2=│EP2│2=(1+t2)y22,所以1m2+n12+│2q(│p-mq·)n=(1+t2y)21·+(yy221y2)2+(1+t22()p│-qqy)1y2│=│q│...     

Robbery

1.Mr. Peter always takes a 9:53 train to leave the city where he works on Friday morning and two hours later,he will get to his villa(['vil(e)]别墅)in the suburb ( [' sΛb(e):b ] 郊区 ).But one Friday,he took a night train to his villa.