VTMO 2016 — Grade 5 — final

What is the hundreds digit of $1\times2\times3\times4\times5\times6\times7\times8\times9\times10$?

A $10$-question multiple-choice section awards $+ 2$ correct, $- 1$ incorrect and $0$ blank. Van scores $12$ and leaves one blank. How many are correct?

A quantity of apples exactly fills $10$ small $+ 8$ large boxes or $5$ small $+ 10$ large boxes. How many small boxes alone are needed?

Stack one red, two blue and two yellow boxes: equal colors are not adjacent; neither top nor bottom is yellow; the middle is not red; a yellow is immediately below red. Find the bottom-to-top order.

An, Binh, Chau have $53$ candies total. An eats $4$, Binh eats $\frac{2}{5}$ of his, and Chau receives $7$. Then An=Binh and Chau=An+Binh. Find Chau’s initial count.

Squares of sides $6 \,\mathrm{cm}$ and $2 \,\mathrm{cm}$ are joined as shown. Find the black triangular area.

$60$ students each enroll in $3$ classes. Each class has at most $15$ students and one teacher; each teacher teaches at most $2$ classes. Find the minimum teachers.

At $14 : 00$ Tom writes $12$. Every $5$ minutes Jerry replaces it with the product of its two digits plus $12$. For example, at $14 : 05$ it is $14$. What number appears at $15 : 30$?

Distinct digits $A , B$ appear in the long multiplication $\overline{AB}\times\overline{BA}$ as shown. Each partial product has three digits; the final product has four digits and ends in $3$. Find the final product.

There are $12$ rows of $10$ seats and $108$ students. What is the greatest number of rows with different numbers of seated students?

A pomelo basket weighs three times an orange basket. Removing $4 \,\mathrm{kg}$ from each makes pomelos four times oranges. Find both initial weights.

Luong does not teach on Saturdays or Sundays. In December $2016$ there are no three consecutive days off. What is the minimum number of teaching days?