VTMO 2026 - Grade 5 - Final - SET 6 - Self-study

Thanh overlaps two identical rulers by $10$ cm as shown. Their combined end-to-end length is then $50$ cm. How many centimetres should the rulers overlap so that the combined length is $56$ cm?

Towns A and B are $180$ km apart. One car travels from A to B in 3 hours 36 minutes and immediately returns towards A. Another car travels from B to A in 4 hours 30 minutes and immediately returns towards B. They start at the same time and meet for the second time at C. How far is C from A? Show your reasoning.

Practice convention: Each car travels at its own constant speed on the same route in both directions.

Mother buys green and red balloons. She gives some to Bình and all the rest to Châu. Châu gets two more balloons than Bình. Bình gets three more red balloons than the number of green balloons Châu gets. How many more red balloons than green balloons did Mother buy?

The diagram shows a $4\times7$ grid. How many square blocks have a sum of entries divisible by $3$? Do not count $1\times1$ squares.

Trees are planted along one side of a road, $20$ m apart. There are 28 trees, including one at each end. Thirteen streetlights are also placed at equal intervals along the same road, with a streetlight at each end. How far apart are consecutive streetlights?

In rectangle ABCD, $DC=60$ cm and $BC=50$ cm. Diagonals AC and BD intersect at their common midpoint O. Point E is on AD with $DE=\frac13DA$. EC meets BD at N, and EB meets AC at M. Find the total area of the two shaded triangles DNC and AMB, in square centimetres. Show your reasoning.

The two sequences of figures follow the patterns shown. The first four figures in the first sequence contain $1,3,6,10$ circles, and those in the second contain $1,4,9,16$ circles.
An has more than 800 but fewer than 1400 circles. Using every circle, An can make a figure in the first sequence. Using every circle again, An can also make a figure in the second sequence. How many circles does An have?

A bag contains 5 green, 9 red and 3 yellow marbles. Some white marbles are added. The white and yellow marbles together then make up $\frac57$ of all the marbles. How many white marbles are added?

What is the hundreds digit of $1\times2\times3\times4\times5\times6\times7\times8\times9\times10$?

Taps A and B fill an empty tank as follows:
• Together they fill it in 30 minutes.
• Running A for 12 minutes, then shutting A and running B for 42 minutes, also fills it.
How many minutes before B should A be opened so that the tank becomes full with equal amounts supplied by the two taps? Show your reasoning.

Practice convention: Each tap has a constant flow rate whenever it is open.

In 2024, someone born in 1997 is 27 years old, while the sum of the digits of 1997 is 26. These are consecutive natural numbers. In 2024, how many birth years have the property that the age and the sum of the birth-year digits are consecutive natural numbers? Use age equal to 2024 minus the birth year, as in the example.

Among birth years from 1900 to 2021 inclusive, how many have age equal to the sum of the digits of the birth year? Here age means $2021-\text{birth year}$.

Each different symbol represents a different number. The total of each row is shown on its right. Find the values represented by the square and the heart. Give the square first and the heart second.