VTMO 2026 - Grade 6 - Semifinal - SET 8 - Self-study

The square has side length $6$ cm. Its top and left sides are each split into lengths $2$ cm and $4$ cm as shown. What is the ratio of the shaded area to the area of the square?

Việt has a special bicycle: the rear wheel has diameter $60$ cm and the front wheel has diameter $50$ cm. On a journey, the rear wheel makes $700$ revolutions. How many revolutions does the front wheel make? Assume that both wheels roll without slipping.

Minh draws the Venn diagram below to show the relationships among trapezoids, isosceles trapezoids, parallelograms and rhombuses, but forgets the labels. Which set does the shaded region represent?

Convention used in this diagram: a trapezoid has at least one pair of parallel sides; an isosceles trapezoid has equal angles at each base (so rectangles are included).

What is the sum of the digits of $9999999992$?

Four rectangular wooden blocks, each measuring $1\text{ cm}\times1\text{ cm}\times3\text{ cm}$, are arranged into each of the five solids shown below. Which solid has the greatest total surface area?

Use the numbers $1,2,3,\ldots,32$ exactly once each as numerators and denominators to form $16$ fractions. At most how many of these fractions can have integer values?

Three circles of radii $7$ cm, $8$ cm and $9$ cm are externally tangent to one another as shown. Find the perimeter, in centimetres, of the triangle whose vertices are their centres.

The long division below divides the four-digit number $\overline{V100}$ by the two-digit number $\overline{VT}$, giving the two-digit quotient $\overline{MO}$ with remainder $0$. Different letters represent different digits. Find $\overline{VIET}+\overline{VTMO}$. The two layouts show the same division.

In the grid, each number outside the $3\times3$ square is the sum of the three entries in its row or column. Find $a+c+c$.

A standard die has six faces, and the numbers on each pair of opposite faces sum to $7$. It is rolled along the arrows, without lifting it or turning it in place. At its final position, the three visible faces have $a,b,c$ dots as shown. Find $a+b+c$.

A square is divided into four equal smaller squares, and the process is repeated in the upper-left square as shown. Find the ratio of the shaded area in the displayed figure to the area of the original square.

A teacher distributes $70$ candies among $6$ students. Each student receives at least one candy, and no two students receive the same number. What is the smallest possible number received by the student who gets the most candies?

Let $A$ be the greatest four-digit odd number whose digits are all different. Find the sum of its digits.

Let $x,y$ be natural numbers less than $10$ satisfying $2x^2-y^2=1$. How many ordered pairs $(x,y)$ are there?

A sequence starts with $1$ and $3$. From the third term onward, a term in an odd-numbered position is the preceding term minus $1$, and a term in an even-numbered position is twice the preceding term. What is the $12$th term?

In square $PQRS$, $M$ and $N$ are the midpoints of $PS$ and $RS$, respectively. Triangle $SMN$ has area $18\text{ cm}^2$. Find the area of triangle $QMN$.

Twenty students take a quiz. No two students answer the same number of questions correctly, and each question is answered correctly by at most three students. What is the least possible number of questions in the quiz?

Châu writes a four-digit number and asks Dũng to guess it.

Dũng: “Is it 2896?”
Châu: “Exactly one digit is correct, and it is in the correct position.”
Dũng: “Is it 5273?”
Châu: “Exactly two digits are correct, but both are in the wrong positions.”
Dũng: “Is it 1702?”
Châu: “All four digits are correct, but all four are in the wrong positions.”

What number did Châu write?

In the twenty-first century, from $2001$ through $2100$ inclusive, what is the greatest number of consecutive years in which January $13$ is not a Friday?

The figures follow the pattern shown below. Which of the following numbers cannot be the number of shaded unit squares in a figure in this sequence?