VTMO 2026 - Grade 5 - Semifinal - SET 1 - Self-study

One of An, Bình, Châu and Dũng broke Mr Ba’s window while playing football. Mr Ba asked who did it. An said, “Bình did it.” Bình said, “Dũng did it.” Châu and Dũng each said, “I did not do it.” Exactly one of the four children told the truth. Who broke the window?

An and Bình have some cards. If Bình gives An 1 card, An will have twice as many as Bình. If An gives Bình 1 card instead, they will have equal numbers. How many cards does An have?

Seven friends each shake hands with every other friend once. How many handshakes are there?

Pi colours the letters in TITAN EDUCATION. Different letters must have different colours, and identical letters must have the same colour. How many colours are needed?

What is the perimeter of the figure below?

The centre cell of a $3\times3$ grid is removed. Fill the other cells with 1 through 8, each once. The top row totals 9, the bottom row 21, the left column 13 and the right column 15. Find $A+B+C+D$.

A rectangle has perimeter 42 cm and width 4 cm. How many centimetres long is it?

Find $N$.
$N=(140+135+130+125+120+115)-(110+115+120+125+130+135)$.

Mr Nam picks oranges and pomelos. He picks three times as many oranges as pomelos. Which number below could be the total number of fruits?

Every year has at least one Friday the 13th and at most three. For example, 2015 had three, in February, March and November. Starting from 2023, how many more years must pass before the next year with three Friday-the-13th dates?

Quang brings money to buy notebooks. Buying 9 leaves him 8,000 đồng; buying 13 leaves him 40,000 đồng short. How much money did he bring?

A rectangular wooden block is cut along the dashed lines into 8 identical cubes, as shown. The small cubes are rearranged into a large cube. Its total surface area is $360\text{ cm}^2$ less than that of the original block. What is the side length, in centimetres, of the large cube?

Let $A$ be the set of natural numbers whose digits are all different and sum to 18. Let $D$ be the largest number in $A$ minus the smallest. What is the digit sum of $D$?

Mr Tư has 538 oranges in five baskets. He adds 2, 4, 8, 16 and 32 oranges to baskets 1, 2, 3, 4 and 5 respectively. All five baskets then contain equal numbers. How many oranges were originally in basket 1?

Split $\frac{21}{22}$ as $\frac{a}{b}+\frac{c}{d}$, where $\frac{a}{b}$, $\frac{c}{d}$ and $\frac{b}{d}$ are all in lowest terms, and $a,b,c,d$ are nonzero natural numbers. How many different splits are there? Reversing the summands counts as the same split; for example, $\frac{21}{22}=\frac12+\frac5{11}=\frac5{11}+\frac12$ is one split.

Last week, Biển Đông Hotel was closed for repairs on Monday and Tuesday, so rooms could be booked only on the other five days. Saturday had 18 rooms booked, strictly more than any other day; Wednesday had 15, strictly fewer than any other day. Every room was booked exactly four times that week. How many rooms does the hotel have? Explain.

Write one of 0, 1 and 2 in each cell of a $4\times5$ grid so that the sum in every row and every column is divisible by 3. What is the greatest possible number of 1s?

The numbers 229 and 419 are examples of three-digit numbers whose digits have product 36. How many such three-digit numbers are there?

An election has 6,000 votes for two candidates. After 90% of the votes have been counted, candidate A has 53% of the counted votes. What is the smallest number of the remaining votes candidate B must receive to win with more votes than A? Explain.

An, Bình and Châu have 29 candies altogether. An has three times as many as Bình, and Châu has 4 more than Bình. How many candies does An have?