VTMO 2026 - Grade 5 - Final - SET 5 - Self-study

There are 48 blue and red marbles in four boxes A, B, C and D. In A there are equally many red and blue marbles. In B, C and D the red counts are respectively 2, 3 and 6 times the blue counts. The four blue counts are consecutive natural numbers, and the total blue count is 20 less than the total red count. Find the blue and red counts in each box, with reasoning.

Find the area of the shaded region in the diagram, in square centimetres. Show your reasoning.

Two bookcases contain some books. If 6 books are moved from the first to the second, the ratio of books in the first to the second is $1:2$. If instead 3 books are moved from the second to the first, the ratio is $7:5$. How many books are there altogether? Explain.

An writes all consecutive natural numbers from 2024 to $n$ in increasing order. He repeatedly replaces the two largest numbers currently present by their difference until only one number remains. The four-digit number $n$ is divisible by 4. What is the smallest possible $n$ for which the final number is 0?

(a) An takes the average of four natural numbers. The first is 12 more than the average, the second is $2/3$ of the average, the third is 19 and the fourth is 25. Find the sum of the first and second numbers.

(b) Mr An buys boxes of cakes. Type A has 4 cakes per box and costs 60,000 đồng per box; type B has 6 cakes and costs 60,000 đồng; type C has 5 cakes and costs 80,000 đồng. He spends 960,000 đồng on 73 cakes. The number of type C boxes is 9 less than the total number of type A and B boxes. How many type A boxes does he buy? Show your reasoning for both parts.

Calculate $80\times3-70\times4+60\times5-50\times6+40\times7-30\times8+20\times9$.

A pen costs 10 thousand đồng and a notebook costs 15 thousand đồng. An, Bình and Châu buy pens and notebooks as follows:
• An buys 23 pens and 12 notebooks.
• Bình buys 35 items altogether and spends 30 thousand đồng more than An.
• Châu spends more than Bình but buys fewer notebooks than Bình.
What is the smallest possible number of pens Châu buys?

How many triangles of area $4\text{ cm}^2$ can be formed by choosing three of the points shown? Adjacent points in each horizontal or vertical row are $1$ cm apart.

Minh writes the ten natural numbers $1,2,3,4,5,\ldots,9,10$ on a board. Each time, Minh erases any two numbers and replaces them with their sum plus $1$. Minh repeats this until only one number remains. What is that number?

An has $\frac23$ as many candies as Bình. Bình has $\frac34$ as many as Châu, and Châu has $\frac45$ as many as Dũng. Dũng's number of candies is what fraction of An's?

Class IQ5A has 15 pupils. Their teacher gives every pupil some candies. Eight pupils receive more than one candy, five receive more than two, and three receive more than three. Exactly one pupil receives five candies, and nobody receives more than five. How many candies does the teacher give out altogether?

At 14:00, teacher Tom writes $12$ on the board. Every five minutes, Jerry erases the current number and writes the product of its two digits plus $12$. For example, at 14:05 Jerry writes $1\times2+12=14$. What number is on the board at 15:30?

A year is called palindromic if its number reads the same forwards and backwards. For example, 121 is palindromic but 2022 is not. A philosopher was born after the year AD 100. His birth year and death year were both palindromic. Including those two years, his lifetime contained exactly 11 palindromic years. He lived no more than 100 years. In what year did he die?