## Subject 1.

Ionel's mother bought bulbs of tulips, daffodils, and lilies, at least one of each type of flower. Ionel's father bought 3 bulbs of tulips, 6 bulbs of daffodils, and 4 times as many bulbs of lilies as his mother did. Ionel noticed that the number of lily bulbs is three times the number of tulip bulbs, and in total, there are 30 flower bulbs.

Prof. Victor Nicolae, Simion Petre, Bucharest

| Details of the solution | Associated scoring |
| :--- | :---: |
| Let $g, z$, and $l$ be the number of tulip, daffodil, and lily bulbs bought by the mother. The father bought $4l$ lily bulbs. (See the adjacent figure!) The total number of tulip bulbs is $g+3$, the total number of daffodil bulbs is $z+6$, and the total number of lily bulbs is $5l$. | $\mathbf{2 p}$ |
| But $5l = 3(g+3) = 3g + 9$, so the total number of flower bulbs is $(g+3) + (z+6) + (3g+9) = 30$. Therefore, $4g + z = 12$. | $\mathbf{1 p}$ |
| This means that $z$ is divisible by 4. Since $z$ is not equal to 0, it follows that $z \in \{4, 8\}$. | $\mathbf{1 p}$ |
| If $z = 8$, then $g = 1$ and the total number of lily bulbs would be $5l = 3 \cdot 1 + 9 = 12$, which is false. | $\mathbf{1 p}$ |
| Therefore, $z = 4, g = 2$, and $5l = 3 \cdot 2 + 9 = 15$, so $l = 3$. | $\mathbf{1 p}$ |
| The total number of tulip bulbs is 5, the total number of daffodil bulbs is 10, and the total number of lily bulbs is 15. | $\mathbf{1 p}$ |
1. Ionel's mother bought bulbs of tulips, lilies, and roses, at least one of each type of flower. Ionel's father bought 3 more tulip bulbs, 6 more lily bulbs, and 4 times as many rose bulbs as his mother did. Ionel notices that the number of rose bulbs is three times the number of tulip bulbs, and in total, there are 30 flower bulbs. How many bulbs of each type of flower did Ionel's parents buy in total?