Exercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are either diamonds or crystals, and the total value is $5000000 €$. They cut each precious stone in half, and each takes one half of each stone. The total value of Martin's stones is $2000000 €$. In euros, what was the total initial value of the diamonds contained in the treasure?

Only a numerical answer is expected here.
Solution to Exercise 8: Let $r$ be the number of stones that are diamonds, and let $d_{1}, \ldots, d_{r}$ be the respective masses of the diamonds unearthed. We also denote $c$ as the mass of crystal unearthed. We aim to determine the value of $100\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)$.

Since the total value of the precious stones unearthed is 5000000, we have

$$
100\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)+3 c=5000000
$$

Since Martin receives half of the total mass of diamonds and half of the total mass of crystal, and the value of Martin's share is 2000000, we also have

$$
100\left[\left(\frac{d_{1}}{2}\right)^{2}+\ldots+\left(\frac{d_{r}}{2}\right)^{2}\right]+3 \cdot \frac{c}{2}=2000000
$$

By subtracting twice (2) from (1), we get

$$
100\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)-2 \times 25\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)+3 c-2 \times 3 \cdot \frac{c}{2}=5000000-2 \times 2000000
$$

which simplifies to $50\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)=1000000$. Therefore, $100\left(d_{1}^{2}+\ldots+d_{r}^{2}\right)=2000000$, and the initial total value of the diamonds in the treasure is 2000000.

Grader Comments: The exercise was well done by the majority. The mistakes made are mainly due to poor modeling of the problem statement. Several students simply wrote a numerically incorrect result and thus received no points for this first exercise, which was the simplest in the high school subject. Even if only a numerical answer is expected, it is recommended to write down the reasoning to earn some points.