## Subject 4.

Ana, Barbu, and Cristi together have an amount of money less than 100 lei, and each of them has more than 5 lei. Ana and Barbu together have three times more than Cristi, and Barbu and Cristi together have three times more than Ana. Find the largest and smallest amounts of money that the three children can have.

## Note: All subjects are mandatory

Time allowed: 2 hours

## MATHEMATICS OLYMPIAD  LOCAL STAGE  JANUARY 18, 2014

## GRADE 5  Scoring Guidelines
## Subject 4.

Let a, b, c be the three sums

$$
\begin{array}{cl}
a+b+c>5, b>5, c>5 & \ldots \ldots .1 \mathbf{1 p} . \\
a+b=3 c \text { and } b+c=3 a & \ldots \ldots .1 \mathbf{1 p} . \\
\text { by subtracting the two relations we get } a=c & \ldots \ldots .1 \mathbf{1 p} . \\
\text { substituting into one of the previous relations we get } b=2 c \text { (or } b=2 a \text { ) } & \ldots \ldots .1 \mathbf{1 p} . \\
\text { from } a+b+c<100 \text { we get } 4 c<100 \text {, so } c<25 & \ldots \ldots .1 \mathbf{1 p} . \\
\text { the largest sums }: a=24 \text { lei, } b=48 \text { lei, } c=24 \text { lei } & \ldots \ldots .1 \mathbf{1 p} . \\
\text { the smallest sums }: a=6 \text { lei, } b=12 \text { lei, } c=6 \text { lei } & \ldots \ldots .1 \mathbf{1 p}
\end{array}
$$