VTMO 2026 - Grade 5 - Semifinal - SET 7 - Self-study

We can write 8 as a product of natural numbers whose sum is also 8. For example, $8=1\times1\times2\times4$ and $8=1+1+2+4$. How many ways can 20 be written as a product of natural numbers whose sum is also 20?

Practice convention: use at least two factors. Rearranging the same factors does not make a new way.

Two containers hold 70 litres of water altogether. Someone removes 20 litres from the first container and adds to the second an amount equal to the water already in it. The water left in the first container is then twice the water in the second. How many litres were originally in the first container?

The average of 101 consecutive even numbers is 198. What is the sum of the smallest and largest numbers in the sequence?

How many triangles are in the picture below?

The first four figures of a pattern are shown below. How many small squares are in the 2012th figure?

A square has a side equal to the width of a rectangle. The rectangle has perimeter 100 cm and length 40 cm. Find the perimeter of the square, in centimetres.

Numbers are entered in a $5\times5$ square grid. The row sums, in order, are 10, 13, 15, 16 and 17. The first four column sums, in order, are 9, 8, 14 and 25. Find the sum of the fifth column.

Find the sum of the three smallest distinct natural numbers whose digit sum is 24.

Minh’s father asks him to count the cows grazing in a field. Instead of giving the number directly, Minh says that the cows have 315 heads and legs altogether. How many cows are in the field?

Thanh writes all two-digit numbers that contain the digit 5. How many numbers does Thanh write?

In a hall, 70% of the people are men. After 108 men leave, the number of women is three times the number of men remaining. How many people were originally in the hall?

January 13, 2017 was a Friday. In which year below is January 13 NOT a Friday?

Practice adaptation: more than one listed year satisfies the question. Select ALL the correct options.

Mr Binh gives his pupils 276 cards of four types: Naruto, One Piece, Conan and Pokemon. The bar chart shows the numbers of each type, but the bar labels and scale are missing. The horizontal gridlines are equally spaced. There are 46 more Pokemon cards than One Piece cards, and Conan is not the most numerous type. How many Naruto cards does he give out?

Five boxes contain the letters A, B, O, R and V as shown. Some letters are removed from each box until exactly one letter remains in each box, with different letters in different boxes. Which letter remains in box 5?

A teacher writes the numbers 1 to 6 on the board. Each time, pupils choose two different numbers to make a fraction and simplify it to lowest terms. What is the greatest number of distinct fraction values they can make?

The figure below consists of two joined squares. Find the area of the shaded region, in square centimetres.

In the diagram, each $2\times2$ square pattern represents a digit, and each row represents a three-digit number. The four numbers are 913, 895, 628 and 238, in some order. Find B.

Given that $\circledast\,5=5+4+3+2+1$, find $\circledast\,15-\circledast\,10$.

The numbers 1 to 9 are placed in the $3\times3$ grid, one number in each cell, according to these rules:

1, 8 and 6 are in the top row.

8, 3, 5, 1, 9 and 4 are not in the right column.

9, 4 and 2 are in the bottom row.

6, 4, 3, 1, 7 and 2 are not in the left column.
What number is in the middle cell?

A solid is assembled from identical cubes of side 1 cm. Its front-to-back and right-to-left views are shown below. What is the greatest possible total surface area of the solid, in square centimetres? The left picture is the front view; the right picture is the right-side view.