Bözsi and Rózsi are selecting scarves in the store while chatting. Rózsi: Give me 5 tens, I'll buy that pretty one, I'll still have some small change left. - Bözsi: You know, in principle I can lend you no more than 40 Ft. - R: That's too little. - B: Give me 40, I'll buy it, and I'll still have money left for the cinema. - R: I'll never lend more than 30 over 30. - B: Then neither of us can buy it. - Salesperson: This scarf is exactly 20% cheaper. - B: Oh, but it's beautiful, 20 Ft would be enough from you. - R: I also have enough with 3 tens. Give it to me! - B: I couldn't bear it if it were yours. - R: Let's not argue, the white one is 4 Ft cheaper again. - B: That's true, but which one of us will contribute to the other? - R: But you're jealous! I would have exactly enough with one twenty. Of course, then it would only be mine. Let's both take this smaller one, it costs only three-quarters as much as the white one. This way, I would owe you just a small change, because you couldn't give me a ten, right? - B: Okay. - (2 minutes later to the salesperson) B: Nini, we didn't even see that one yet. Please, could you exchange mine for that one! - E: If you add 6 Ft. - B: Isn't this one better on me, Rózsi? But now we only have enough for one of us to take the tram.

How much money did the girls have approximately (that is, what was the minimum and maximum they had), and approximately how expensive were the scarves they bought?
Let Bözsi have $x$ money, Rózsi have $y \mathrm{Ft}$, the price of the first scarf be $s \mathrm{Ft}$, the amount of money deemed sufficient for the cinema be $m \mathrm{Ft}$, and the remaining money of the two girls after shopping (the leftover for the tram) be $v \mathrm{Ft}$. Thus, the price of the second scarf is $0.8 s \mathrm{Ft}$, the price of the white scarf is $0.8 s - 4 \mathrm{Ft}$, the price of the small one is $0.6 s - 3 \mathrm{Ft}$ (which Rózsi bought), and finally, the price of the scarf bought by Bözsi is $0.6 s + 3 \mathrm{Ft}$.

The relationships between the different prices and the girls' money can mostly be expressed with inequalities. Rózsi's money, supplemented by $50 \mathrm{Ft}$, is more than $s$, but with $40 \mathrm{Ft}$ borrowed, it still does not reach $s$:

$$
y + 40 < s < y + 50
$$

From (9) and (2):

$$
0.4 s + 54 + v < s \leq 0.4 s + 64 + v - m
$$

We have no further information about $v$, and there is no relationship to it, so we can only eliminate it with an assumption about its value. Assuming the tram ticket costs 70 fillers, $0.70 \leq v \leq 1.40$, thus the value on the left side of (12) is at least $0.4 s + 54.70$, and the value on the right side is at most $0.4 s + 65.39 - m$. We can only eliminate $m$ in this way. Since the multiplier of $m$ is negative and this term is on the larger side, we must use a lower bound, thus minimizing the reduction of the larger side. Assuming the cinema ticket costs at least $2 \mathrm{Ft}$, i.e., $m \geq 2$, (12) becomes:

$$
0.4 s + 54.70 < s \leq 0.4 s + 63.39
$$

From this, with the already seen steps:

$$
91.16 < s \leq 105.65
$$

From (7) and (9), similarly:

$$
0.4 s + 14.70 \leq 0.4 s + 14 + v < 0.6 s - 3 < 0.4 s + 24 + v < 0.4 s + 25.40
$$

$88.50 < s < 142$.

Finally, from (3) and (9):

$$
s \leq 113.50
$$

So far, we have obtained lower and upper bounds for $s$ in (10), (11), (13), and (14), and an upper bound in (15). The largest lower bound is 91.16 from (13), and the smallest upper bound is 105 from (11), but $s$ cannot be equal to 105. Since 20% of $s$, or one-fifth of $s$, is an exactly payable amount according to the seller, $s$ must be an integer multiple of 0.05. Therefore, (1)-(4), (6), (7), (10), (11), (13)-(15) are all satisfied if and only if:

$$
91.20 \leq s \leq 104.95
$$

From this, for Rózsi's money, using (5a): multiplying by 0.8 and then subtracting 24:

$$
72.96 \leq 0.8 s \leq 83.96 \quad \text{and} \quad 48.96 \leq y \leq 59.96
$$

Similarly, for Bözsi's money, using (9):

$$
61.18 \leq x \leq 67.37
$$

(We considered that the smallest value of $s$ was obtained with $v=0.70$, and the largest value with $v=1.39$, so we used these values for the limits of $x$ calculated from them.)

Finally, for the prices of the scarves, from the expressions we derived for the purchased scarf $z=0.6 s - 3$, and $z+6$:

$$
51.72 \leq z \leq 59.97, \quad \text{and} \quad 52.72 \leq z+6 \leq 65.97
$$

Let's test our results with the smallest and largest values of $s$! Remember that the former was obtained with the assumption $v=0.70$, and the latter with $v=1.39$ and $m=2$.

$$
\begin{array}{lll}
\text{If} & s=91.20 \mathrm{Ft}, & s=104.95 \mathrm{Ft}, \\
\text{then} & y=48.96 \mathrm{Ft}, x=61.18 \mathrm{Ft} & y=59.96 \mathrm{Ft}, x=67.37 \mathrm{Ft},
\end{array}
$$

And indeed, the following inequalities hold:

$$
\begin{aligned}
& 88.96 < 91.20 < 98.96 \\
& 99.96 < 104.95 < 109.96 \\
& 91.18 < 91.20 \leq 99.18 \\
& 97.37 < 104.95 \leq 105.37 \\
& (71.18 <) 72.96 \leq 81.18 \\
& (77.37 <) \quad 83.96 \leq 87.37 \\
& (68.96 <) 72.96 \leq 78.96, \\
& (79.96 <) \quad 83.96 \leq 89.96 \\
& 48.96 < 51.72 < 58.96 \\
& 59.96 < 59.97 < 69.96 \\
& 51.18 < 51.72 < 61.18 \\
& 57.37 < 59.97 < 67.37
\end{aligned}
$$

According to these, Bözsi could pay Rózsi's shortfall of $51.72 - 48.96 = 2.76 \mathrm{Ft}$, and $59.97 - 59.96 = 0.01 \mathrm{Ft}$, plus the $6 \mathrm{Ft}$ surcharge, and finally had 0.70, or $1.39 \mathrm{Ft}$ left.

Katona Mária (Budapest, Szilágyi E. lg. II. o. t.) Szidarovszky Ágnes (Budapest, Ságvári E. gyak. lg. II. o.)

Remarks 1. According to the conversation, the borrowed amount was primarily a multiple of $10 \mathrm{Ft}$, and smaller amounts were only borrowed out of necessity. Thus, it can be assumed that Bözsi would not have had enough $10 \mathrm{Ft}$ to buy the second scarf, and Rózsi would not have had enough $20 \mathrm{Ft}$. In this case, (3) and (4) would be replaced by:

$$
x + 10 < 0.8 s \leq x + 20, \quad \text{and} \quad y + 20 < 0.8 s \leq y + 30
$$

The solution also satisfies this, and the corresponding values are in parentheses in the tests.

2. The rapid conversation suggests quick calculations and rounded values. Thus, it is likely that the scarves cost 100, 80, 76.57, and $63 \mathrm{Ft}$, and the girls had about 65 and $56 \mathrm{Ft}$.