## Task 2 - 190622

In a shelf of a HO retail store, there are six gift items priced at $15 \mathrm{M}, 16 \mathrm{M}, 18 \mathrm{M}, 19 \mathrm{M}, 20 \mathrm{M}$, and $31 \mathrm{M}$, with exactly one piece of each type.

One customer bought exactly two of these gifts, another exactly three. The second customer had to pay twice as much as the first.

Determine from these details which of the six gifts were bought by the first and which by the second customer!
The total price of all six gifts is 119 M. Since exactly five gifts were bought, one was left out. If the one left out was priced at $15 \mathrm{M}, 16 \mathrm{M}, 18 \mathrm{M}, 19 \mathrm{M}, 20 \mathrm{M}$, or $31 \mathrm{M}$, then the two buyers together paid $101 \mathrm{M}, 103 \mathrm{M}, 101 \mathrm{M}, 100 \mathrm{M}, 99 \mathrm{M}$, or 88 M, respectively.

If the first buyer paid $x$ Mark, then the second buyer paid $2 x$ Mark. The total amount paid by both, $3 x$ Mark, must be divisible by 3. This is only true for the amount of $99 \mathrm{M}$.

Therefore, the gift priced at $20 \mathrm{M}$ was not bought; furthermore, $3 x=99$, so the first buyer paid $33 \mathrm{M}$.

If he had bought one of the gifts priced at $16 \mathrm{M}, 19 \mathrm{M}$, or $31 \mathrm{M}$, the other would have had to cost $17 \mathrm{M}, 14 \mathrm{M}$, or $2 \mathrm{M}$, respectively; but these prices did not occur.

Thus, the first buyer bought the gifts priced at $15 \mathrm{M}$ and $18 \mathrm{M}$, and the second buyer consequently bought the gifts priced at $16 \mathrm{M}, 19 \mathrm{M}$, and $31 \mathrm{M}$.