Interesting Earth, Sun and Moon ខែ​មីនា 31, 2009

 (A B C is the intersection point of external tangent line of the each pair of circles)

ចូរអ្នកបង្ហាញថាA ,B, C រត់ត្រង់គ្នា។

Simply prove that A, B , C is collinear.

 
Hint 1 : It is just a geometry problem that applied Menelaus theorem. Menelaus theorem is what? Check wiki. 

Hint 2: You may choose the triangle to apply menelaus to be the center of each three circle.

Set and Subset ខែ​មីនា 30, 2009

S ជាសំនុំនៃ ៨​ ចំនួនគត់វិជ្ជមានខុសគ្នា ហើយចំនួននីមួយៗ សុទ្ឋតែតូចជាង ៣០។ ចូរបង្ហាញថា S  មានសំនុំរងពីរផ្សេងគ្នា ដែល ផលបូកធាតុទាំងអស់នៅក្នុងសំនុំរងទាំងពិរស្មើគ្នា។

ឧទាហរណ៏  S= { 2 , 4, 5 8,12 15,28 , 24}  សំនុំរងទាំងពីរអាចជា {2,4,12,15}​​       និង   {4 , 5, 24}    ផលបូក=៣៣។

Let S be a set of eight positive integers each of which is less than 30. Show that there must be two distinct subsets of S whose elements add up to the same sum. For instance, if the eight numbers are {2,4,5,8,12,15,18,24} , the two distinct subsets can be {2,4,12,15} and {4 , 5, 24}. The sum of both of these is 33.

Hint:

How many subset that S can produce? What are the range of sum of elements in these subsets range from?

Solution