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

A garden is square. If the length of each side decreases by $10\%$, by what percentage does its area decrease?

Thanh has $1024$ coins in his pocket. Each time, he takes out half of the coins still in his pocket. How many times must he do this to leave exactly one coin?

The homes of An, Binh, Hanh and Phuc and their school lie along the same road, in the order shown. Phuc lives $600\text{ m}$ from school. One day all four start at the same time, and each takes $15$ minutes to reach school at a constant speed. When Phuc has walked $\frac13$ of his route, Hanh reaches Phuc’s home, Binh reaches Hanh’s home and An reaches Binh’s home. How far is An’s home from school, in metres?

A football team plays $90$ matches in a season. It wins $40$ of its first $50$ matches. How many of the remaining matches must it win to finish with wins in exactly $70\%$ of the season’s matches?

A shop sells a new kind of cake. On the first day, it sells $200$ cakes and makes a $20\%$ loss. On the second day, it sells $500$ cakes and makes a profit of $10$ million dong. It makes the same number of cakes each day and discards any unsold cakes. What is the selling price per cake, in thousands of dong?

Assumptions for this calculation: the total daily production cost and the selling price per cake are unchanged, and the $20\%$ loss is measured against production cost.

1 January 2015 was a Thursday. What day of the week was the last day of January 2015?

During Black Friday, Hai’s mother buys him a watch. With a $40\%$ discount, she has $16$ money units left. At the regular price, she would be short by $4$ money units. What is the regular price, in the same money units?

Quang starts with a square of area $256\text{ cm}^2$ (Figure 1). He adds four squares at its four corners, each with one quarter of its area (Figure 2). Next, at the three outward corners of each newly added square, he adds three more squares, each having one quarter of that square’s area (Figure 3). He repeats this rule to obtain Figure 4 and continues in the same way. What is the total area of all the shaded squares in Figure 6, in square centimetres?

A natural number is called perfect if the sum of all its positive divisors other than itself equals the number. For example, the divisors of $6$ are $1,2,3,6$, and $1+2+3=6$, so $6$ is perfect. Which of the following numbers is perfect?

Ti climbs a staircase, taking either one or two steps at a time. In how many different ways can he climb exactly $8$ steps?

Mother walks around a lake, completing one lap in $12$ minutes. Binh roller-skates in the same direction and overtakes her once every $18$ minutes. If Binh instead skates in the opposite direction, how many minutes pass between consecutive meetings? Give a full solution.

Assumption: both keep the same constant speeds in either direction.

A teacher gives candies to the students of class 6C in order. The first student receives $1$ candy and then $\frac1{21}$ of the candies remaining. The second receives $2$ candies and then $\frac1{21}$ of the remainder; the third receives $3$ candies and then $\frac1{21}$ of the remainder; and so on. The last student takes all the candies left, and this number equals that student’s position in the order. Each student receives the same total number of candies. How many students receive candies?

Class 6A has $16$ students, including $6$ girls. The number of girls is what percentage of the number of boys?

Find $x$:

$(x+2)-(2x+3)+(3x+4)-(4x+5)+\cdots+(19x+20)-(20x+21)=100$.

A machine processes a natural number repeatedly. If the current number is even, it divides it by $2$; if it is odd, it subtracts $1$. It stops when the result reaches $0$. What is the smallest starting number that reaches $0$ after exactly $10$ operations? For example, $6\to3\to2\to1\to0$ takes four operations.

Sixty chairs are arranged around a circular table. Students sit so that each seated student has exactly one other student immediately next to them. What is the greatest possible number of seated students?

Three spinners are shown. Spinner A has the eight integers $1,2,3,4,-1,-2,-3,-4$; spinner B has the two operations $+$ and $-$; spinner C has the same eight integers as A. Viet spins A, then B, then C, and calculates the number from A, followed by the operation from B, followed by the number from C. For example, obtaining $4,-,2$ gives $4-2=2$. How many different ordered spinner outcomes give a positive integer?

The diagram is a net of a cube. Its faces are labelled $1,5,6,x,y,z$. Both the number $5$ and the number $x$ equal the average of the numbers on the four faces adjacent to their own face. Find $x$.

This year, Chau’s age and her mother’s age add up to $50$ years. In $17$ years, Chau will be as old as her mother was when Chau was born. How old is Chau this year?

The diagram shows a net of a cube with a different symbol on each face. Which face is opposite the diamond-shaped symbol?