VTMO 2026 - Grade 6 - Semifinal - SET 9 - Self-study

Equilateral triangle $ABC$ is made up of rhombus $DEFB$ and equilateral triangles $ADE$ and $ECF$, as shown. Given $DE=3$ cm, find the length of $BC$, in centimetres.

A square has side length $14$ cm. A circle is drawn inside it, and a smaller square is drawn inside the circle, as shown. Find the shaded area.

Let $A=1+2+2^2+2^3+\cdots+2^{2014}+2^{2015}$. Find the remainder when $A$ is divided by $31$.

In square $PQRS$, $M$ and $N$ are the midpoints of $PS$ and $RS$, respectively. Triangle $SMN$ has area $18$ cm². Find the area of triangle $QMN$, in cm².

Three consecutive odd numbers are all prime. What is their sum?

Given $$\frac ab=\frac{1\frac 12\times 2\frac 23\times 3\frac 34\times\cdots\times 98\frac{98}{99}\times 99\frac{99}{100}}{1\times 2\times 3\times\cdots\times 99\times 100},$$ where $a/b$ is in lowest terms with positive integers $a,b$, find $a+b$.

The diagram shows a game with three pegs and four discs $A,B,C,D$. Binh removes one disc at a time until all four discs have been removed. How many different removal orders are possible? A disc is lifted off the top of its peg.

Sets $C$ and $D$ are shown in the Venn diagram. Which statement is correct?

Chau and Phuong each want to buy a book. Chau has $3/4$ of its original price, and Phuong has $1/2$ of that price. If the price of each book is reduced by $30\,000$, their combined money buys exactly two books. What is the original price of one book?

A digital clock displays HH:MM, from 00:00 to 23:59. For example, at 8:05 a.m. it displays 08:05, and at 9:09 p.m. it displays 21:09. How many times in one day does its display contain exactly the four digits 2, 0, 2, 0, in any order?

Two positive integers have sum $1323$ and greatest common divisor $147$. How many pairs of integers satisfy these conditions? Count each pair only once, regardless of order.

The product of two positive integers is $10\,000$, and neither integer is divisible by $10$. Find their positive difference.

A teacher chooses two numbers from 1,2,3,4,5,6. He tells Phong their product and Sang their sum.

Phong: “I know the product, but I cannot determine the two numbers.”

Sang: “I know the sum, but I cannot determine the two numbers either.”

Phong: “Then I now know the two numbers.”

What is their sum?

Model clarification: the two chosen numbers are distinct. Both pupils hear the whole dialogue, speak truthfully, make correct deductions and use all earlier statements.

Chau lies on Friday, Saturday and Sunday, and tells the truth on the other days. Minh tells the truth on Tuesday, Wednesday and Thursday, and lies on the other days. On which day can both say, “Tomorrow I will lie”?

What is the sum of the digits of the number $9999999992$ (nine digits $9$ followed by $2$)?

Pi cuts one sheet of paper into 10 pieces. He then takes one piece and cuts it into 10 smaller pieces. He repeats this last action two more times. How many pieces does he have in total?

Different letters represent different digits in the following multiplications: $$\overline{ABCDEF}\times 2=\overline{CDEFAB},$$ $$\overline{ABCDEF}\times 3=\overline{BCDEFA},$$ $$\overline{ABCDEF}\times 4=\overline{EFABCD}.$$ Find the four-digit number $\overline{ABCD}$.

How many three-digit numbers are divisible by 8?

A sequence starts $1,2,3$. From the fourth term onward, each new term is obtained from the previous three by adding the first two and subtracting the third. The sequence begins $1,2,3,0,5,-2,7,-4,\ldots$. Find the sum of its first $2016$ terms.

There are 65 pupils in a hall, each wearing either a white shirt or a blue shirt, not both colours. There are 25 boys in white shirts,23 pupils in blue shirts and 28 girls. How many girls are wearing blue shirts?