VTMO 2026 - Grade 5 - Final - SET 2 - Self-study

A “strange number” has five digits, with nonzero first and last digits. Reading either from left to right or from right to left, the first digit is divisible by 1, the second by 2, the third by 3, the fourth by 4 and the fifth by 5. For example, 54685 is a strange number. How many strange numbers are there?

A teacher divides a bag of candies equally among the 10 students of class IQ5. If each girl gives back 1 candy and each boy receives 1 extra candy, 4 candies remain. If one girl is absent, the number of boys is unchanged, and that girl’s original share is divided equally among the other girls, each of those girls receives 1 extra candy. How many candies are in the bag?

In a certain year there are more Sundays than Tuesdays, and more Saturdays than Thursdays. On what day of the week does 17 December fall that year?

The figure is formed from a square of side $10$ cm and a rectangle of length $15$ cm. Find the difference between the unshaded area and the shaded area, in $\mathrm{cm}^2$.

In rectangle $ABCD$, take $M$ on $AB$ and $O$ on $MC$ so that $MO:OC=1:3$. Line $DO$ meets $AB$ at $N$. The two shaded triangles have total area $10\ \mathrm{cm}^2$. Find the area of rectangle $ABCD$, in $\mathrm{cm}^2$, and explain.

Dũng’s father is packing apples into small and large boxes. Either 10 small boxes and 8 large boxes, or 5 small boxes and 10 large boxes, hold exactly all the apples. How many small boxes would hold all the apples if only small boxes were used?

Teams A and B each have 10 workers. Working alone, A builds a road in 14 days and a bridge in 25 days; B builds the same road in 18 days and the same bridge in 15 days. Team C consists of 5 workers from A and 5 from B. It finishes the road and then builds the bridge. How many days does C need in total? Show your reasoning.

Practice clarification: for each job, workers within each original team have equal, constant productivity, and their work rates add. Workers from different teams need not have equal productivity.

Calculate $(9\times8\times7+6-5)\times4+3+2-1$.

An, Bình and Châu always lie. Each owns one stone, either blue or red. An says, “My stone is the same colour as Bình’s.” Bình says, “My stone is the same colour as Châu’s.” Châu says, “Exactly two of us have red stones.” How many of these statements are true?

• An has a blue stone.

• Bình has a blue stone.

• Châu has a red stone.

• An and Châu have stones of different colours.

A square cloth has one corner folded in as shown. The original square’s area and the remaining area after folding are consecutive even natural numbers. Find the original area, in $\mathrm{cm}^2$.

Practice clarification: the crease joins the midpoints of two adjacent sides, and the folded corner reaches the square’s centre. The remaining area is the visible footprint, with the overlap counted once.

Find the value of
$$\dfrac{3}{2+\dfrac{1}{2+\dfrac{1}{100+\dfrac{1}{101+\dfrac{1}{102}}}}}+\dfrac{3}{1+\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{100+\dfrac{1}{101+\dfrac{1}{102}}}}}}.$$

Five girls stand in a row. The first three girls have a total age of 33 years; the middle three, 30 years; and the last three, 35 years. The first girl is 12 and the last girl is 15. How old is the second girl from the front?

Minh visits his grandparents and sees one basket of oranges and one of pomelos. His grandfather says, “The pomelo basket currently weighs three times as much as the orange basket. If I remove 4 kg from each basket, the pomelo basket will weigh four times as much as the orange basket.” Find the original weight of each basket and explain.