Lesson Math Olympiad level problem on caves and bats

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Math Olympiad level problem on caves and bats

Problem 1

There is a mountain with 30 bat caves in a row that contain 340 bats, in all.
Any 7 caves in a row contain exactly 77 bats. Suppose the first cave has 7 times as many bats as the last cave.
How many bats are in the 29th cave?

Solution

Let me start my solution from the reference to an Internet dictionary

"in a row" means "in succession".

In this problem, it is the only meanings of this term.

1.  Since any 7 caves in a row contain exactly 77 bats, we have, in particular

         = 77,   (1)

         = 77.   (2)


    Subtract equation (1) from equation (2).  The terms from  to  will cancel each other, and you will get

         -  = 0,   or

         = .


    Making the same step by step, you will get

         = ,
         = ,
         = ,
         = ,
         = ,
         = ,
         =  = 

    and so on. It means that the sequence  , , , . . . ,  is PERIODICAL with the period 7.


    It is the FIRST IDEA of the solution.



2.  Its consequence is that   = .


    Therefore, from the condition, we have this equation

         =  = .     (3)


    It is the SECOND IDEA of the solution.



3.  From the condition, we have 

         = 77,

         = 77,

         = 77,

         = 77.

    since any 7 caves in a row contain exactly 77 bats.


    Add these 4 equation (both sides). You will get

         +  +  + . . . +  = 4*77 = 308.


    It means that  

         +  = 340 - 308 = 32.    (4)


    It is the THIRD IDEA of the solution.



4.  Now, from equations (3) and (4) we have

         +  = 32.


    It implies   = 32,   or


         = 32/8 = 4.


    Hence,   =  = 7*4 = 28.


ANSWER.   = 28.  The 29-th cave has 28 bats.