VTMO 2013 — Grade 5 — final

Find a two-digit natural number such that inserting $0$ between its digits makes a number seven times the original.

An has $\frac{2}{3}$ as many candies as Binh; Binh $\frac{3}{4}$ as many as Chau; Chau $\frac{4}{5}$ as many as Dung. What is Dung’s count divided by An’s?

Two $15 \,\mathrm{cm}$ rulers retain only marks $3$ and $4$ and overlap as pictured: mark $3$ of one coincides with $4$ of the other. Find their total span $L$.

A box contains cards numbered $1$ to $7$. The first person randomly takes $3$ cards, and the second takes $2$ of the remaining cards. The first person knows for certain that the second person’s sum is even. What is the sum of the first person’s cards?

A children’s group has mean age $11$. Adding Nam, age $17$, raises the mean to $12$. How many were originally in the group?

A square plot’s dimensions are increased by $3 m$ and $5 m$, raising its area by $87 m^{2}$. Find its original area.

Friends pick mushrooms. If An gives Binh half of An’s mushrooms, all friends have equal counts. If instead An gives Chau all of An’s mushrooms, Chau has as many as all the others together. How many friends are there?

$A$ and $B$ are $180 \,\mathrm{km}$ apart. One car takes $3 h = 36$ min from A to $B$ and immediately returns; another takes $4 h = 30$ min from $B$ to $A$ and returns. They start together and meet a second time at $C$. How far is $C$ from $A$?