Question 1166355
.



            The solution by tutor  @solver91311  is not correct.

            To demonstrate it,  I will construct a contradictory example.



<pre>
Let all the books are 15 cm wide.


Then the total number of books is  {{{240/15}}} = 16,

and we can place  ONLY  3  such books at each shelf.



So, having 5 shelves, we can place only  3*5 = 15 such books, and we need then

the 6-th shelf for the 16-th book.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Now, after completing this counter-example, &nbsp;I can solve the problem in full.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My statement is that &nbsp;6 &nbsp;shelves is always enough.



<pre>
1)  In the 1-st shelf,  I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



2)  In the 2-nd shelf,  I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



3)  In the 3-rd shelf,  I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



4)  In the 4-th shelf,  I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



5)  In the 5-th shelf,  I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



6)  In the 6-th shelf I can fill at least 40 cm of 56 cm.


    Indeed, if less than 40 cm is filled, then I can add any book (since it is no 
    thicker than 16 cm).



So, I can fill at least 40 cm of 56 cm in each of 6 shelves.



Taken together,  6 times 40 cm comprise  2 m 40 cm,

which means that ALL the books will be placed in 6 shelves.
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;***********************************
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;THE PROOF IS COMPLETED.
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;***********************************



It is a &nbsp;TRUE &nbsp;Math &nbsp;Olympiad level &nbsp;Math &nbsp;problem &nbsp;(!)