Limit: Intermediate Level ^_^

លំហាត់លីមីត៖ សម្រាប់សិស្សពូកែ

46th Canada MO: Problem Number 1

ប្រធានលំហាត់៖

គេឲ្យ $ a_1,\ a_2,\ ...,\ a_n$ ជាបណ្តាចំនួនពិតវិជ្ជមាន ដែលមានផលគុណស្មើ 1។ 

បង្ហាញថាផលបូកៈ

\[\dfrac{a_1}{1+a_1}+\dfrac{a_2}{(1+a_1)(1+a_2)}+\dfrac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+...+\dfrac{a_n}{(1+a_1)(1+a_2)...(1+a_n)}\]

គឺធំជាង រឺ ស្មើនឹង $\dfrac{2^n-1}{2^n}$ ។

ដំណោះស្រាយ

$​​​\dfrac{a_m}{(1+a_1)(1+a_2)...(1+a_m)}$

$=\dfrac{1+a_m}{(1+a_1)(1+a_2)...(1+a_m)}-\dfrac{1}{(1+a_1)(1+a_2)...(1+a_m)}$

$=\dfrac{1}{(1+a_1)...(1+a_{m-1})}-\dfrac{1}{(1+a_1)...(1+a_m)}$

ដូចនេះ បើយើងយក $b_j=(1+a_1)(1+a_2)...(1+a_j)$, ដែល $b_0=0$, ពេលនោះយើងបានផលបូក

\[\sum_{j=1}^{n}\dfrac{a_j}{(1+a_1)...(1+a_j)}=\sum_{j=1}^{n}(\dfrac{1}{b_j-1}-\dfrac{1}{b_j})=1-\dfrac{1}{b_n}\]

យកចិត្តទុកដាក់ថា $b_n=(1+a_1)(1+a_2)...(1+a_n)\ge(2\sqrt{a_1})(2\sqrt{a_2})...(2\sqrt{a_n})=2^n$

ដែល សមភាពកើតមានលុះត្រាតែគ្រប់ $a_i$ ស្មើនឹង $1$ ។ ដូចនេះ,

\[1-\dfrac{1}{b_n}\ge 1-\dfrac{1}{2^n}=\dfrac{2^n-1}{2^n}\]

ដើម្បីដឹងថាតម្លៃតូចបំផុតនេះ អាចកើតមានរឺទេ, គឺយើងជំនួនគ្រប់ $a_i=1$ ចូល យើងបានៈ

\[\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^n}=\dfrac{2^{n-1}+2^{n-2}+...+1}{2^n}=\dfrac{2^n-1}{2^n}\]

បញ្ហាត្រូវបានស្រាយបញ្ជាក់។

Physics Exercise #24: VNPhO 30-4-2004 Grade 10, P3

Let enjoy new exercise today! For all high school students especially physics lovers ^_^

Free Physics Book: A Guide to Physics Problems Part 1

A Guide to Physics Problems Part 1: Mechanics, Relativity, and Electrodynamics by Sidney B.Cahn and Boris E.Nadgorny is a book full of good exercise for high school students who preparing for outstanding students and for 1st year student in physics department.

Hope this book will make you feel love physics more!

 About this book:

+Tittle:  3000 Solved Problems in Physics

+Size: 6.y5MB

+File Type: PDF
+Language: English

+Link Download: Dropbox File

Download More English Physics Book Here:
http://highschoolcam.blogspot.com/p/english-physics-books.html

Free Maths Book: Mathematics Problems Collection For Outstanding Students Competition

គណិតវិទ្យាជ្រើសរើសសម្រាប់ត្រៀមប្រឡងសិស្ស ពូកែជាសៀវភៅដែលរៀបរៀងដោយអតីតជ័យលាភី ទូទាំងប្រទេសឆ្នាំ២០១១ ចំណាត់ថ្នាក់លេខ៥។

សៀវភៅនេះ  រួមមាន៧៤លំហាត់ ត្រូវបានជ្រើសរើស ដោយសម្រិតសម្រាំង សម្រាប់ត្រៀមប្រឡងសិស្ស ពូកែថ្នាក់ខេត្ត និងថ្នាក់ជាតិ។

សូមអរគុណចំពោះប្អូនប្រុស សុខ ពិសី ជាអ្នករៀប រៀងសៀវភៅនេះ និងបានផ្ញើរមកកាន់ខ្ញុំផ្ទាល់​ដើម្បីធ្វើ ការចែកចាយបន្ត ដល់ប្អូនៗជំនាន់ក្រោយ។

 About this book:

+Tittle: គណិតវិទ្យាជ្រើសរើសត្រៀមសិស្សពូកែ
+Language: Khmer 

+Size: 1.72MB

+File Type: PDF

+Link Download: Dropbox File



ទាញយកសៀវភៅជាច្រើនទៀតនៅទីនេះ ៖

• សៀវភៅគណិតជាភាសាខ្មែរ៖ http://highschoolcam.blogspot.com/p/khmer-maths-books.html
• សៀវភៅគណិតជាភាសាអង់គ្លេស៖ http://highschoolcam.blogspot.com/p/english-maths-books.html
• សៀវភៅគណិតជាភាសាវៀតណាម៖ http://highschoolcam.blogspot.com/p/vietnam-maths-books.html

Free Maths Book: Geometry-4th-edition _ Schaums'outline

Geometry by Schaum's oulines is a good geometry book high school students.

This book include geometry in plan theorem and problems with full solution good for all high school student prepare for outstanding student test and use as fundamental for geometry studying for APMO and IMO test.

Hope this book will be good for all of you who like mathematics. Let enjoy download it for free.

 About this book:

+Tittle:  Geometry-4th-edition _ Schaums'outline
+Language: English

+Size: 3.85MB

+File Type: PDF

+Link Download: Dropbox File

Download More Book Here:
http://highschoolcam.blogspot.com/p/english-maths-books.html

Free Maths Test Collection: APMO 1989-2014 Full Test Paper With Full Some Full Solutions

         ដើម្បីរួមចំនែកក្នុងការស្វែងរកឯកសារត្រៀមប្រឡង គណិវិទ្យាអូឡាំព្យាដអាស៊ីប៉ាស៊ីភិក នៅឆ្នាំក្រោយ របស់ប្អូនៗ ខ្ញុំបានប្រមូលផ្តុំរាល់វិញ្ញាសាប្រឡង និងអត្រាកំនែនៅបណ្តាឆ្នាំមុនៗ ក្នុងការប្រឡងនេះ  ចងក្រងទៅជាសៀវភៅមួយក្បាលនេះ។

         សង្ឃឹមថាឯកសារមួយនេះ  នឹងចូលរួមចំនែកយ៉ាងសកម្ម ក្នុងការពង្រឹង និងពង្រឺកបន្ថែម ទៀតលើមុខវិជ្ជាគណិតវិទ្យារបស់ប្អូនៗ  ជាពិសេសសម្រាប់អ្នកដែលត្រៀមខ្លួនក្នុងការប្រកួត ប្រជែងលក្ខណៈអន្តរជាតិមួយនេះ។

          ប្អូនៗ និងមិត្តអ្នកសិក្សាអាចចូលទៅទាញយកឯកសារមួយនេះ តាមរយៈតំនភ្ជាប់ខាងក្រោមៈ
                       
                      https://www.dropbox.com/s/ar4obgynjn7juov/APMO%201989-2014.pdf

Free Physics Book: 3000 Solved Problems in Physics

3000 Solved Problems in Physics is a good book for foundation year students and also for high school students to prepare for outstanding students test.

For this book it not full yet, i will try to find out the full version one and will be update for u all as soon as.

Hope this book will be good for all of you who like mathematics. Let enjoy download it for free.

    About this book:

+Tittle:  3000 Solved Problems in Physics

+Size: 7.71MB

+File Type: PDF

+Link Download: Dropbox File

Download More Book Here:
http://highschoolcam.blogspot.com/p/english-physics-books.html

Free Physics Book: Physics for scientists and engineers 5th edition (Instructor's Solution Manual)

Physics for scientists and engineers 5th edition (Instructor's Solution Manual)

Big thank to brother Leav Menghuy for this book.


Hope this book will be good for all of you who like mathematics. Let enjoy download it for free.


    About this book:

+Tittle:  Physics for scientists and engineers 5th edition (Instructor's Solution Manual)

+Size: 20.0MB

+File Type: PDF

+Link Download: Dropbox File

Test Paper: Cambodia Physics Olympiad Nation Round 2014 For Grade 12

Sorry for this document that it not so clear, please what for new clear version one. 

I will update it when i got a new one. Big thank to brother Ham Senghong for this test paper! 

Test result: Nation Round 2014 Result Was Released This Evening!

Congratulation for all little sisters and little brother who has pass the nation round, all of u did a great job!! 

I'm so proud with all of u ^_^ 

"Physics Champion Grade 12"

"Maths Champion Grade 12"

Free Maths Book: International-Mathematical-Olympiad-1959-1999

 

    International Mathematical Olympiad 1959-1999
by sir.ISTVAN REIMAN is one of the good book for
high school students who want to joint outstanding  students test, especially IMO test.


    Hope this book will be good for all of you who like mathematics. Let enjoy download it for free.


    About this book:

+Tittle:  Tuyen Tap Bat Dang Thuc

+Size: 1.84MB

+File Type: PDF

+Link Download: Dropbox File

Test Paper: Hot Hot!!, Unofficial Test Paper of KhMO 2014 Had Release

So now we got what we wait to see so far, KhMO 2014 Grade 12 Full Unofficial Test Paper.

Big thank to brother Mengsay for this test paper~ We will wait to see the official test that will release soon.

Free Physics Book: Mechanics For Foundation Year (Hang Sim)

សៀវភៅរូបវិទ្យាមួយក្បាលនេះ  គឺត្រូវបានប្រើ
ប្រាស់សម្រាប់និស្សិតនៃមហាវិទ្យារូបវិទ្យា ក្នុងសកលវិទ្យាភូមិន្ទភ្នំពេញ។  ដោយឡែក ក្នុង
សៀវភៅនេះដែរ ក៏មានលំហាត់មួយចំនួនធំ អាច
ប្រើប្រាស់បានចំពោះសិស្សកំរិតវិទ្យាល័យ ជាពិសេសសម្រាប់បំប៉នសិស្សពូកែរូបវិទ្យា។


សៀវភៅមេកានិចសំរាប់ថ្នាក់ឆ្នាំមូលដ្ឋាននេះ ត្រូវបានរៀបរៀងដោយលោកគ្រូ អនុបណ្ឌិត ហង់ ស៊ីម ​បង្រៀននៅសាកលវិទ្យាល័យ ភូមិន្ទភ្នំពេញ។

About this book:
+Source: http://eebookforu.wordpress.com/

+Tittle:  Mechanic For Foundation Year Student

+Size: 67.7MB 

+File Type: PDF

+Link Download: Click Here

Free Maths Book: Inequality collection in Maths Forum (VN language)

For who like to use VN maths book to study and prepare for outstanding student test it good to use this book to study inequality. It a good book that have nearly all kind of inequality in high school maths, was collected by a group of VN teacher and VN outstanding student in their maths forum.

Hope u can use it as a useful documents in your studying.

About this book:

+Tittle:  Tuyen Tap Bat Dang Thuc

+Size: 1.84MB

+File Type: PDF

+Link Download: Dropbox File

Test Paper: VN Team Selection Test 2014

Again in this year, we don't have the name in official IMO 2014 web. But for VN they already prepare for it every year now, and they just finished with the name of the team this year with updated exercise more difficult than last year. May be we just can sit and take a look at other country to show their talented student in international test like IMO. 

I always hope for one day ours country will be change and become strong and better than this, that why i'm trying my best to help the next generation the my way like this. 

You, next generation, please don't lost hope and must try to fight harder and keep waiting for one day that your time will come and show your skill with other country!! May be next year, IMO will be hold in Thailand, we may be can joint ? I'm not sure yet, but since it near ours country then i think we will can joint it!  Good luck and keep fighting! 

Test Paper: APMO 2014 Original Test Paper With Full Solution Has Released

ទើបតែទទួលបានថ្មីៗ នូវវិញ្ញាសាដើម និងដំណោះស្រាយពេញលេញនៃការប្រឡងគណិតវិទ្យាអាស៊ីប៉ាស៊ីភិក ឆ្នាំ២០១៤។

តើអ្នកដោះស្រាយវិញ្ញាសានេះបានល្អទេ?  មានសង្ឃឹមថានឹងទទួលបានមេដាយសំរាប់ប្រទេសខ្មែរយើងទេ? សង្ឃឹមថាឆ្នាំនេះ  អាចនឹងមានបេក្ខជនដែលទទួលបានលទ្ទផលល្អជាងឆ្នាំកន្លង បើទោះជាឆ្នាំនេះ វិញ្ញាសាមានលក្ខណៈលំបាកជាងឆ្នាំទៅក៏ដោយ។

ជួនពរឲ្យអ្នកដែលចូលប្រឡង APMO នៅឆ្នាំនេះទទួលបានលទ្ធផលល្អ។

អ្នកអាចទាញយកលំហាត់ និងដំណោះស្រាយក្នុងការប្រឡងនេះ តាមរយៈតំណភ្ជាប់ខាងក្រោម៖

https://www.dropbox.com/s/15tfb5ngr0uxi4w/APMO%202014%20Full%20Solution.pdf

Circuit Lesson: Load in the circuit ^_^

 Canteen like crucial nodes making me and my friends closer together. "Money From Father" represents the source exploited throughoutly be students. The current (money) supplied by "money from father" won't pass through friends, it is poured directly into my wallet. The amount of money can be divided into many parts where "Girl Friend" takes an enormous quantity. Ironically, girl friends are considered as Load, ....
!!! So what does it mean by "Load" !!!! ?? ^_^

Test Paper: KhMO 2010 Nation Round Full Test Paper+Solution

Get this document by click on this link:
                  https://www.dropbox.com/s/wdelriijp12dvg0/KhMO%20Nation%20Round.pdf