Problem 5. The Stojanovski family has three sons: Nikola, Riste, and Aleksandar. The father, Petar Stojanovski, is 35 years old. The sum of the ages of the three children is half the age of the mother, Elena, who is 5 years younger than the father, Petar. Last year, the oldest son was as old as the sum of the ages of his brothers.

How old is the mother, Elena? How old is the oldest son, Aleksandar? If Nikola is three times younger than Riste, how old is the youngest of the brothers?
Solution. First method. Let $a, b, c, d$ and $e$ be the ages of Nikola, Riste, and Aleksandar, father Petar, and mother Elena, respectively. It follows that

$$
d=35, e=30, 2(a+b+c)=30, c-1=a-1+b-1 \text { and } b=3 a
$$

It follows that

$$
a+b+c=15 \text { and } a+b-c=1 \text {, i.e., } a+b-c+2 c=15
$$

or $1+2 c=15$ or $c=7$. From this, $a+b=8$ and $b=3 a$, from which we have that $a=2$ and $b=6$.

Thus, Nikola is 2, Riste is 6, Aleksandar is 7, Petar is 35, and Elena is 30 years old.

Second method. Father Petar is 35 years old. Mother Elena is 5 years younger, i.e., 30 years old. The sum of the children's ages is half the age of the mother and amounts to 15. One year ago, the sum of the children's ages was 12 years. Then the oldest son Aleksandar was half the sum, i.e., 6 years old. It follows that Aleksandar is 7 years old and Nikola and Riste together are 8 years old. Since Nikola is 3 times younger than Riste, Nikola is 2, and Riste is 6 years old.

## VI section - nine-year-old