## Task 4 - 190714

Six students helped with the fruit harvest. They received recognition bonuses according to their performance. Each of them donated half of the received amount to the solidarity fund. Furthermore, the following is known about these students:

(1) None of them donated less than $6 \mathrm{M}$ and none more than $12 \mathrm{M}$.

(2) Konrad donated more than Peter.

(3) Helga donated more than Gisela, Gisela more than Peter, Peter more than Inge.

(4) Frank donated more than Helga and Helga more than Konrad.

(5) Helga donated $2 \mathrm{M}$ less than Frank, Peter $2 \mathrm{M}$ more than Inge.

(6) All donated full mark amounts.

How much money did each of the students receive for picking fruit?
Let $f, g, h, i, k, p$ be the amounts of money donated by Frank, Gisela, Helga, Inga, Konrad, and Peter, respectively, in Marks. From the statements in the problem, we have:

(7) $h > g > p, f > h > k > p > i$ from (2), (3), (4)

(8) $f = h + 2, p = i + 2$ from (5)

(9) $f \leq 12, i \geq 6$ from (1)

(10) $h \leq 10, p \geq 8$ from (8)

(11) If in (9) even $f = 6$, then from (8) $h = 8$;

However, under the condition (10), no integer $g$ can satisfy $h > g > p$. Therefore, this case is excluded,

(12) i.e., in (9) $f = 12, i = 6$ must hold.

(13) Thus, from (8) $h = 10, p = 8$;

(14) According to this, the inequalities $h > g > p$ and $h > k > p$ can only be satisfied by integers $g = 9, k = 9$.

Since the donations thus determined $[(12), (13), (14)]$ were half of the received amounts, it follows: Frank received $24 \mathrm{M}$, Gisela $18 \mathrm{M}$, Helga $20 \mathrm{M}$, Inga $12 \mathrm{M}$, Konrad $18 \mathrm{M}$, and Peter $16 \mathrm{M}$.

Solutions from the 1st Round 1979 adopted from [5]

### 4.21.2 2nd Round 1979, Class 7