## Task 3 - 090613

In a competition of the mathematical student magazine "alpha", the four given geometric figures were to be assigned the correct names.

In exactly $\frac{3}{25}$ of the submitted solutions, all four given figures were assigned the correct names. In exactly twice as many solutions, two figures were assigned the correct names and two figures were assigned the wrong names.

The number of solutions with exactly three incorrect assignments was exactly four times the number of correct solutions. Exactly 240 of the submitted solutions contained no correct assignments. No further submissions were received.

Determine the total number of solutions submitted for this competition!
The number of submitted solutions is $n$. Then it holds:

Exactly four are correct, i.e., $\frac{3}{25} n$, exactly two are correct, i.e., $\frac{6}{25} n$, exactly one is correct, i.e., $4 \cdot \frac{3}{25} n$, and no solution is correct are 240. The sum is

$$
n=\frac{3}{25} n+\frac{6}{25} n+4 \cdot \frac{3}{25} n+240=\frac{21}{25} n+240 \quad \rightarrow \quad \frac{4}{25} n=240
$$

Thus, there were a total of $n=240 \cdot \frac{25}{4}=1500$ submissions.