27 = 3 x 3 x 3

45 = 3 x 3 x 5

63 = 3 x 3 x 7

75 = 3 x 5 x 5

99 = 3 x 3 x 11

kick me

Let SS be the sum of all the mnmn numbers in the table. Note that after an operation, each number stay the same or turns to its negative. Hence there are at most 2mn2mn tables. So SS can only have finitely many possible values. To make the sum of the numbers in each line nonnegative, just look for a line whose numbers have a negative sum. If no such line exists, then we are done. Otherwise, reverse the sign of all the numbers in the line. Then SS increases. Since SS has finitely many possible values, SS can increase finitely many times. So eventually the sum of the numbers in every line must be nonnegative. 

Pigeonhole principle ?

Each time, a person takes 1, 2, 3, or 4 chips. The strategy for a player to win the game is taking the number of chips so that the number of chips left is the form of 5k + 1 .

If It is my turn, and there are 2014 chips in the pile, I shoud take 3 chips and the left is 2011 (=5.k +1). And if the opponent takes x chips (x = 1, 2, 3, 4) , i win take 5 - x chips to guarantee the left is always in the form 5k + 1. And finally, the opponent will take the last chip.

 1
 

0.3 

Click on me

it is 9999

8 number 

240 cakes 

kích mình nhé

240 cakes 

This number is 99 =  This number is ninety-nine

That number is 99 = That number is ninety-nine

chọn ý nào cũng được

that number 9999

  that number 99