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, 2010

Algebra Difficulty 5.6 AIME, harder Prove it Ukraine

Vitalii, Michael and Olexandr were given $n$ dollars by their mother each one. Vitalii had to spend all his money for buying books, Michael for buying notebooks, Olexandr for pens. Vitalii had bought 1 book, Michael 2 notebooks and Olexandr 5 pens. After that, they together had $n$ dollars. Prove that one of the boys can buy one more item.

Solution

Assuming the contrary. Denote by $q_1, q_2$ and $q_3$ the costs of one book, notebook and pen, respectively, and by $r_1, r_2, r_3$ the change our boys were given. Obviously $r_1 < q_1, r_2 < q_2, r_3 < q_3$ . From the statement we have
$n = q_1 + r_1, \quad n = 2q_2 + r_2, \quad n = 5q_3 + r_3$
and we can write the next inequalities: $n = q_1 + r_1 > 2r_1$ , $n = 2q_2 + r_2 > 3r_2$ , $n = 5q_3 + r_3 > 6r_3 \Rightarrow r_1 < \frac{1}{2}n, r_2 < \frac{1}{3}n, r_3 < \frac{1}{6}n$ . Adding them we will have
$n = r_1 + r_2 + r_3 < \frac{n}{2} + \frac{n}{3} + \frac{n}{6} = n$
which is impossible. This contradiction completes the proof.

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