VTMO 2026 - Grade 5 - Final - SET 3 - Self-study

A palindrome reads the same from left to right as from right to left. For example, 1111, 1221 and 12321 are palindromes. How many four-digit palindromes can be formed using the digits 3, 5, 7 and 0? Digits may be used more than once.

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. To win with more votes than A, what is the least number of the remaining votes that candidate B must receive? Show your reasoning.

KenKen is a Japanese arithmetic puzzle. The $5\times5$ grid is divided into cages by thick lines. Each cage is labelled with a target number and an operation.

Examples: $7+$ means the numbers in the cage sum to $7$; $2-$ means the difference of its two numbers is $2$; $2\div$ means the quotient of its two numbers is $2$; $4\times$ means the product of all its numbers is $4$.

Fill the grid with the numbers $1$ to $5$ so that each row and each column contains each number once, and every cage satisfies its label. Find $A+B$.

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 $\mathrm{mm}^2$.

A three-digit number has digit sum 25 and digit product 567. What is the greatest three-digit number satisfying both conditions?

In the multiplication shown, different letters stand for different digits. Each dash is one unknown digit. What is the product?

Find a two-digit natural number such that inserting the digit 0 between its two digits produces a three-digit number seven times the original number.

The pictures show three views of the same die. Write each pair of letters on opposite faces.

Two squares are joined as shown, with side lengths 6 cm and 2 cm. Find the area of the shaded region in square centimetres.

Two taps running together fill an empty tank in 10 minutes. If the first runs for 3 minutes and the second for 9 minutes, they supply $7/10$ of the tank. How many seconds before the first tap should the second be opened so that, by the time the initially empty tank is full, the second has supplied five times as much water as the first?

Exactly one of An, Bình and Châu always tells the truth. An says, “Bình does not always tell the truth.” Bình says, “I do not always tell the truth.” Châu says, “An does not always tell the truth.” At least one of these three statements is false. Who always tells the truth?

Five of the six numbered pieces are used to build the T-shaped solid shown. Which numbered piece is not used?

Nine balls are labelled 1 through 9. An, Bình, Châu and Dũng each take two balls, with no ball used twice. An’s labels sum to 10; Bình’s differ by 1; Châu’s multiply to 24; and the larger of Dũng’s labels divided by the smaller equals 3. Which label is left? Explain your reasoning.