VTMO 2013 — Grade 6 — final

Delete one digit from a five-digit positive integer to obtain a four-digit integer. Their sum is $52713$. Find the original.

Saw $a = 4 m$ log into $20 \,\mathrm{cm}$ pieces. Each cut takes $8$ minutes, with $2$ minutes rest between cuts. Starting $7 am$, when is the work finished?

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?

Find positive integers $a , b$ such that $b$ divides $a + 1 , a = 2 b + 5$, and $a + 7 b$ is prime.

In triangle $ABC , M$ and $N$ are midpoints of $BC$ and $AC$, and $AM$ meets $BN$ at $O$. $a$)Compare the areas of $AON$ and $BOM$. $b$)Find $\frac{OA}{OM}$.