VTMO 2019 — Grade 5 — final

Place $1 \text{--} 13$ once around $13$ circles so neighboring numbers differ by less than $3$. Positions $6 , x , y$ are shown. Find $x + y$.

Five girls stand in a line. First three ages total $33$, middle three $30$, last three $35$. First is $12$, last $15$. Find the second girl’s age.

$60 \%$ of Thanh’s shirts have short sleeves, the rest long. One-third of short-sleeved and one-quarter of long-sleeved shirts are pink, total $9$ pink shirts. Find all shirts.

A square is partitioned into rectangles as shown. The black rectangles’ perimeters total $48 \,\mathrm{cm}$. Find the square’s area.

Mother divides all red and green balloons between Binh and Chau. Chau has two more balloons. Binh’s red count is three more than Chau’s green count. How many more red than green balloons were bought?

A hotel accepts bookings on five days, closing Monday and Tuesday. Saturday has $18$ rooms booked, strictly most; Wednesday $15$, strictly least; each room is booked exactly $4$ times that week. How many rooms are there?

Exactly one of An, Binh, Chau always tells the truth. An says “Binh does not always tell the truth”; Binh says “I do not always tell the truth”; Chau says “An does not always tell the truth”. At least one of these statements is false. Who always tells the truth?

In rectangle $ABCD , M$ lies on $AB , O$ on $MC$ with $MO : OC = 1 : 3$, and $DO$ meets $AB$ at $N$. The two shaded regions total $10 \,\mathrm{cm}^{2}$. Find the rectangle’s area.