VTMO 2026 - Grade 5 - Semifinal - SET 11 - Self-study

A fraction reduces to $2/3$. Its numerator and denominator have a sum of 105. What is its numerator?

The five-digit number $\overline{2X34Y}$ is divisible by 6. What is $Y$ when this number is as small as possible?

There are 60 chairs around a circular table. Pupils sit down so that each seated pupil has exactly one other pupil in the two neighbouring chairs. What is the greatest possible number of seated pupils?

Four discs A, B, C and D are on three pegs as shown. Bình removes one disc at a time until all four are removed. In how many different orders can he do this?

Which expression below has value 17?

Original choices: A. $4\times5+6\times3$; B. $4\times5+6\div3$; C. $6\div3+5\times4$; D. $3\times4\div6+5$; E. $3+4\times5\times6$.

Practice adaptation: enter the matching letter, or NONE if none of the original choices equals 17.

The objects are placed on balance scales as shown. Which object is the heaviest?

Every morning An leaves home for school at 6:30 am. He spends 4 hours 45 minutes studying at school, then walks home and arrives at noon. He walks to school at 100 m/min and returns at 80 m/min. How far is his home from school?

Different letters stand for different digits in the addition shown. Given $A=7$, what is $H$?
$\overline{MATH}+2024=\overline{VTMO}$.

A palindrome reads the same forwards and backwards. For example, 1221 and 181 are palindromes, but 122 is not. How many three-digit palindromes are divisible by 9?

A mother is now 47 years old. In 3 years, she will be twice as old as her child. How old is the child now?

Pupils stand in a line. Pi is in the middle of the whole line and also halfway between An and Bình. An is third from the front; Bình is eight positions behind An. How many pupils are in the line?

The numbers 2 and 0 are written in the alternating pattern shown. Let $S_n$ be the sum of all entries in a chosen $n\times n$ square. For example, the indicated $3\times3$ square has $S_3=10$. An chooses a $10\times10$ square. Find $S_{10}$.

An ant starts at vertex A of the rectangular box shown, travels along its edges, and returns to A. It visits all 8 vertices, visiting each other vertex exactly once before returning to A. What is the shortest possible distance travelled?

Pi repeatedly writes OLYMPICTOANTITAN to form the string
OLYMPICTOANTITANOLYMPICTOANTITAN…
What is the 2014th letter?

Nine cards are numbered 120, 220, 320, 420, 520, 620, 720, 820 and 920. Anh removes two cards and the average of the remaining numbers stays the same as before. Bình then removes two more cards, and the remaining average again stays unchanged. What is the sum of the four numbers removed?

A father is now 36 years old. His age 10 years ago equals the sum of his two children’s present ages. The older child is 6 years older than the younger. How old is the older child now?

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

There are 65 pupils in a hall, each wearing either a white or a blue shirt. There are 25 boys in white shirts, 23 pupils in blue shirts, and 28 girls in total. How many girls wear blue shirts?

Khang writes the consecutive odd numbers starting from 1, with no gaps:
13579111315…
He notices two adjacent digits 9. What is the least number of odd numbers he could have written?

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. Which friend always tells the truth? Explain.