Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:

- Alina, your age is now double the age of Valentina.

a) Show that this could never have happened before and that, after the next birthday of Valentina, it will never happen again.

Then, Alina, who was very clever, exclaimed:

- Dad, exactly 10 years ago, the age of one of us three was double the age of another, and in 10 years, the same fact will happen again!

b) It is known that the oldest of the daughters is over 30 years old. How old is Civela?
Solution

a) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).

If Valentina's age today is $\alpha$, then Alina's age must be $2 \alpha$, so the age difference between the two is given by:

$$
d=2 \alpha-\alpha
$$

We conclude that $d=\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.

Such a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.

b) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.

It is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:

(i) $B$ is double $A$.

(ii) $C$ is double $B$.

(iii) $C$ is double $A$.

By the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:

- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.
- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.
- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.

Observe that the ages of $A$ in cases (i), (ii), and (iii) are

$$
x, \quad y-x \quad \text { and } \quad x+y
$$

respectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations

$$
y-x+10=x \quad \text { and } \quad x+10=x+y
$$

Solving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.

We know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that

$$
x+10=y-x \quad \text { and } \quad y-x+10=x+y .
$$

Solving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to

$$
y-x=15, \quad y=20 \quad \text { and } \quad 2 y=40
$$

Since Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. Therefore, Civela is 15 years old today.