Mária and Anna's age sum is 44 years. Mária is twice as old as Anna was when Mária was half as old as Anna will be when Anna is three times as old as Mária was when Mária was three times as old as Anna. How old are Mária and Anna?
I. solution: The apple tree method leads to a very simple system of equations with 5 unknowns. Currently
$t$ years ago
$u \quad \ll$ then $x-u$
$v \ll$ then $\quad x-v$

| $y$ | years old, |
| :--- | :--- |
| $y-t$ | years old, |
| $y-u$ | years old, |
| $y-v$ | years old, |

Maria

$x$

Anna

According to the problem

$$
\begin{aligned}
x+y & =44 \\
x & =2(y-t) \\
2(x-t) & =y+u \\
y+u & =3(x-v) \\
x-v & =3(y-v)
\end{aligned}
$$

Gradually eliminating the unknowns $v, u, t, y$:
From (4) and (5)
From (6) and (3)
$2 u=9 x-11 y$,
From (7) and (2)
$4 t=9 y-5 x$
finally
$5 y-3 x=0$,
From (8) and (1)
$8 x=220$,

from which

$$
x=27 \frac{1}{2} \text { years, thus } y=16 \frac{1}{2} \text { years. }
$$

From (7)

$$
t=\frac{9 y-5 x}{4}=\frac{11}{4}=2 \frac{3}{4} \text { years }
$$

From (6)

$$
u=\frac{9 x-11 y}{2}=\frac{66}{2}=33 \text { years, }
$$

From (5)

$$
v=\frac{3 y-x}{2}=\frac{22}{2}=11 \text { years }
$$

Tihanyi István (Szeged, Organic Chemistry and Technology II. class)

II. solution: We can also reach the goal with one unknown if we reason backwards from the last time point.

Let Anna be $x$ years old at the time when Maria is $3 x$ years old. The constant age difference is therefore $2 x$ years. When Maria was half as old as Anna will be at $3 \cdot 3 x=9 x$ years old, Maria was $4.5 x$ and Anna was $4.5 x-2 x=2.5 x$ years old. According to the problem, Maria is currently $2 \cdot 2.5 x=5 x$ years old, and thus Anna is $5 x-2 x=3 x$ years old.

Therefore

$$
5 x+3 x=44
$$

from which

$$
x=\frac{44}{8}=5.5
$$

and thus currently Maria is $5 x=27.5$ years old and Anna is $3 x=16.5$ years old.

Kurovszky Zsuzsanna (Esztergom, Dobó Katalin High School II. class)