4. On the trip, there were 4 boys and 4 girls. It is known that each boy is a brother to one girl, i.e., there are four pairs of brothers and sisters from four different families, and the names and surnames of the girls and boys are known respectively. On the trip, they took 32 cakes, and: Maria took 1 cake, Jovanka took 2 cakes, Ivanka took 3 cakes, and Ana took 4 cakes. The boys took: Petreski as many as his sister, Marcoski twice as many as his sister, Poposki three times as many as his sister, and Pavloski four times as many as his sister. What are the names and surnames of the girls?
Solution. Let the girls with the surnames Petreska, Marcoska, Popeska, and Pavloska take $a, b, c$, and $d$ cakes respectively. Then the brother and sister Petreski took $2a$ cakes, Marcoski $3b$ cakes, Poposki $4c$ cakes, and Pavloski $5d$ cakes. Moreover, $a, b, c, d \leq 4$ and $2a + 3b + 4c + 5d = 32$, from which it follows that $b$ and $d$ are both even or both odd numbers. Therefore, we have four possibilities: $b=1, d=3; b=3, d=1; b=2, d=4$ and $b=4, d=2$. It is easily verified that only the case $b=4, d=2, a=3, c=1$ is possible.

Therefore, the girls are: Ivanka Petreska, Ana Marcoska, Maria Popeska, and Jovanka Pavloska.

## 2nd year