58th Ukrainian National Mathematical Olympiad

Question

It is given that the bookshelf can fit 9 of the same thick books, but the 10th one will not fit anymore. Similarly, it can hold 15 of the same thin books, but 16th will not fit anymore. Is it possible for that shelf to hold simultaneously: a) 6 thick and 5 thin books? b) 7 thick and 5 thin books?

Accepted Answer

Let us re-write the statement of the problem as follows. Let us denote the length of the shelf by

S

, the width of the thick book by

x

and the width of the thin book by

y

. Then, we have the conditions:

9 x \leq S < 10 x and 15 y \leq S < 16 y \Leftrightarrow \frac{1}{10} S < x \leq \frac{1}{9} S and \frac{1}{16} S < y \leq \frac{1}{15} S .

a) Let's see what is the maximum width of the space these books would take:

6 x + 5 y \leq \frac{6}{9} S + \frac{5}{15} S = S .

b) Analogously, we show that this set of books cannot fit on the shelf:

7 x + 5 y > \frac{7}{10} S + \frac{5}{16} S = \frac{56 + 25}{80} S > S .