# Monthly Archives: ខែសីហា 2009

## តោះ!លេងហ្គេមម្តង!!

Caesar III: Build a Better Rome | Real-Time Strategy (Seira online) game | 88 MB

Emperor Caesar has assigned you to one of Rome’s provinces and it’s up to you to create a safe, pleasant city that will attract immigrants. Watch your city grow and evolve, and maintain its economy to keep the favor of your citizens… as well as Caesar. Play your cards right and you could become Emperor of Rome itself!

Game Features:
* Build, rule and defend on one screen – no more switching between city, province and battle screens!
* Use the City Construction Kit to build the perfect city, or climb the ladder of Roman politics.
* Now your citizens can give you a glimpse into the life of the common man – talk to them for clues about how to improve your city.
* Appease 5 Gods with temples and festivals. Each God has its own sphere of influence: Please Ceres, and your crops will thrive but dishonour Neptune, and watch your trade ships sink into the sea…
* Various new structures and challenges arise throughout your Governor’s career, for tremendous depth of play and replayabiltiy.
* A truly intuitive, helpful interface lets you jump right in and start building!

System requirements
Windows 95/98/98SE/ME/xp/Vista
Pentium 933 or better
128MB of RAM
32MB graphics/video card

បើចូលចិត្តប្រាប់ផង!!!

The SAP name has been synonymous with state-of-the-art analytical methods since its introduction over 30 years ago. SAP2000 follows in the same tradition featuring a very sophisticated, intuitive and versatile user interface powered by an unmatched analysis engine and design tools for engineers working on transportation, industrial, public works, sports, and other facilities.

From its 3D object based graphical modeling environment to the wide variety of analysis and design options completely integrated across one powerful user interface, SAP2000 has proven to be the most integrated, productive and practical general purpose structural program on the market today.

This intuitive interface allows you to create structural models rapidly and intuitively without long learning curve delays. Now you can harness the power of SAP2000 for all of your analysis and design tasks, including small day-to-day problems. Complex Models can be generated and meshed with powerful Templates built into the interface.

The Advanced Analytical Techniques allow for Step-by-Step Large Deformation Analysis, Multiple P-Delta, Eigen and Ritz Analyses, Cable Analysis, Tension or Compression Only Analysis, Buckling Analysis, Blast Analysis, Fast Nonlinear Analysis for Dampers, Base Isolators and Support Plasticity, Energy Methods for Drift Control and Segmental Construction Analysis.

Bridge Designers can use SAP2000 Bridge Templates for generating Bridge Models, Automated Bridge Live Load Analysis and Design, Bridge Base Isolation, Bridge Construction Sequence Analysis, Large Deformation Cable Supported Bridge Analysis and Pushover Analysis.

SAP2000 is for everyone! SAP2000 is for every project! From a simple small 2D static frame analysis to a large complex 3D nonlinear dynamic analysis, SAP2000 is the answer to all structural analysis and design needs.

BricsCad v9 is the smart choice for architects, engineers,designers-virtually any professional who creates or uses CAD drawings.BricsCad provides unrivaled compatibility with Autodesk AutoCAD and is fully programmable with hundreds of third-party solutions. AutoCAD command line Most CAD professionals spend years mastering AutoCAD commands until they’re second nature.Since these commands help create the majority of CAD work,BricsCad transparently maps them to BricsCad equivalent commands.

DWG COMPATIBILITY
– Native DWG 2007 file format
– Support for AutoCAD versions 2.5 to 2008
– Extended raster image support
– User data file manager
– New LISP engine
– Full SDS support
– New highly compatible COM API
– Support for Object ARX

PRODUCTIVITY
– Powerful Drawing Explorer
– Dockable Properties Bar
– Consolidated Settings Manager
– Customize dialog and editor
– Full ACIS (3D solids) modeling
– Visual Basic for Applications (VBA)

Enjoy!!!

## ប្រយ័ត្នច្រឡំ!!!

គណនា$\mathop {\lim }\limits_{x \to \infty } {{\sin x} \over x}$

សំរាយ

$\forall x \in \Re _ +$យើងមាន$- 1 \le \sin x \le 1$

ដូចនេះ$- {1 \over x} \le {{\sin x} \over x} \le {1 \over x}$

$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \left( { - {1 \over x}} \right) \le \mathop {\lim }\limits_{x \to \infty } {{\sin x} \over x} \le \mathop {\lim }\limits_{x \to \infty } {1 \over x}$

តែ$\mathop {\lim }\limits_{x \to \infty } \left( { - {1 \over x}} \right) = \mathop {\lim }\limits_{x \to \infty } {1 \over x} = 0$

ដូច្នេះ$\mathop {\lim }\limits_{x \to \infty } {{\sin x} \over x} = 0$

ខ្មាសគេមិនស្ទើរដោយសារលំហាត់នឹង​ ទៅជាស្រាយថាបាន១អីណា៎ យាប់ម៉ង

## ធរណីមាត្រម្តង!!!

គេអោយចតុកោណកែង$ABCD$មួយដែល$BC=3AB$$P$និង$Q$ជាពីរចំនុចនៃជ្រុង$[BC]$ដែល$BP=PQ=QC$។ចូរ​បង្ហាញថា$\angle DQC = \angle DBC + \angle DPC$

ដំណោះស្រាយ

បង្ហាញថា$\angle DQC = \angle DBC + \angle DPC$

តាង$\angle DBC= \alpha ; \angle DPC = \beta ; \angle DQC = \lambda$

យើងមាន$\tan \alpha = {1 \over 3} , \tan \beta = {1 \over 2} , \tan \lambda= 1$

ដោយ$\tan \left( {\alpha + \beta } \right) = {{\tan \alpha + \tan \beta } \over {1 - \tan \alpha \tan \beta }} = {{{1 \over 2} + {1 \over 3}} \over {1 - {1 \over 6}}} = 1 = \tan \lambda$

ដូចនេះ$\alpha +\beta = \lambda$$\angle DQC = \angle DBC + \angle DPC$

## លំហាត់កំសាន្ត!!!

១.គេអោយបីចំនួនពិតវិជ្ជមាន$a,b$និត$c$។​ចូរបង្ហាញថា$a^2 + b^2 + c^2 \ge ab + bc + ca$

ដំណោះស្រាយ

បង្ហាញថា$a^2 + b^2 + c^2 \ge ab + bc + ca$

តាមវិសមភាព$AM-GM$គេមាន៖

${{a^2 + b^2 } \over 2} \ge ab$​ (1)

${{b^2 + c^2 } \over 2} \ge bc$ (2)

${{c^2 + a^2 } \over 2} \ge ca$ (3)

បូកវិសមភាព (1), (2) និង ​(3) អង្គនឹងអង្គគេបាន៖

${{a^2 + b^2 + b^2 + c^2 + c^2 + a^2 } \over 2} \ge ab + bc + ca$

ដូចនេះ$a^2 + b^2 + c^2 \ge ab + bc + ca$

២.​ គេអោយចំនួនពិតវិជ្ជមាន$a$និង$b$។ ចូរបង្ហាញថា$\left( {1 + a} \right)\left( {1 + b} \right) \ge \left( {1 + \sqrt {ab} } \right)^2$

អនុវត្តន៍ រកតំលៃតូចបំផុតនៃអនុគមន៍៖

$f\left( x \right) = \left( {1 + 4^{\sin ^2 x} } \right)\left( {1 + 4^{\cos ^2 x} } \right)$ដែល$x \in \Re$

ដំណោះស្រាយ

បង្ហាញថា$\left( {1 + a} \right)\left( {1 + b} \right) \ge \left( {1 + \sqrt {ab} } \right)^2$

យើងមាន$\left( {1 + a} \right)\left( {1 + b} \right) = 1 + \left( {a + b} \right) + ab$(1)

តាមវិសមភាព$AM-GM$គេមាន$a + b \ge 2\sqrt {ab}$

តាម(1)គេទាញបាន$\left( {1 + a} \right)\left( {1 + b} \right) = 1 + 2\sqrt {ab} + ab$

ដោយ$1 + 2\sqrt {ab} + ab = \left( {1 + \sqrt {ab} } \right)^2$

ដូចនេះ$\left( {1 + a} \right)\left( {1 + b} \right) \ge \left( {1 + \sqrt {ab} } \right)^2$

ដោយប្រើវិសមភាពខាងលើយើងទាញបាន៖

$f\left( x \right) = \left( {1 + 4^{\sin ^2 x} } \right)\left( {1 + 4^{\cos ^2 x} } \right) \ge \left( {1 + \sqrt {4^{\sin ^2 x + \cos ^2 x} } } \right)^2 = 9$

ដូចនេះតំលៃអប្បបរិមានៃអនុគមន៍នេះស្មើនឹង$9$

## លំហាត់ពិសេស!!!

េគអោយពហុធាដឹក្រេទី៤៖

$p\left( x \right) = 2009x^4 + 2004x^3 + 2007x^2 + 2003x + 2005$ស្រាយបញ្ជាក់ថា$p\left( x \right) > 0$គ្រប់$x \in \mathbb{R}$

ដំណោះស្រាយ

-ស្រាយបញ្ជាក់ថា$p\left( x \right) > 0$គ្រប់$x \in \mathbb{R}$

+ករណី$x \geqslant 0$នោះយើងឃើញថា$p\left( x \right) \ge 2005 \ge 0$ពិត។

+ករណី$x < 0$

យើងមាន

$p\left( x \right) = 2009x^4 + 2004x^3 + 2007x^2 + 2003x + 2005$

$p\left( x \right) = 5x^4+2004\left( {x^4 + x^3 + x^2 + x + 1} \right) +3x^2-x+1$

$p\left( x \right) = 5x^4+3x^2 + \left( { - x} \right)+1+2004\left( {x^4+x^3+x^2+x+1} \right)$

ចំពោះកន្សោម$x^4+ x^3+ x^2+ x+1$

$= \left( {x^4+ x^3+{{x^2 } \over 4}} \right) + \left( {{{x^2 } \over 4} + x + 1} \right) + {{x^2 } \over 2}$

$= x^2 \left( {x + {1 \over 2}} \right)^2+ \left( {{x \over 2} + 1} \right)^2+ {{x^2 } \over 2} > 0$

នាំអោយ$p\left( x \right) > 0$ពិត

ដូចនេះ$p\left( x \right) > 0$គ្រប់$x \in \mathbb{R}$