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

The operation is illustrated by $\otimes 5=5+4+3+2+1$. Find $\otimes 15-\otimes 10$.

Phong and Khang are discussing an old book.

Phong: “I added all the numbers used to number the pages and got 2024.”
Khang: “That is impossible. Have you made a mistake?”
Phong: “I forgot to say that the book is numbered only from page 3 to the last page, and the numbers on two consecutive pages have faded, so I did not add those two numbers.”

What is the sum of the two faded page numbers?

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

Five runners, An, Bình, Châu, Dũng and Giang, finish a race in distinct positions. Two spectators predict the order from first to fifth:

First spectator: Giang, Bình, Dũng, An, Châu.
Second spectator: An, Châu, Bình, Giang, Dũng.

The first spectator gets exactly three positions correct and the second gets exactly two correct. In which position does Dũng finish?

A plot of land is a square. Increasing one dimension by 3 m and the other by 5 m produces a rectangle whose area is 87 square metres greater. Find the area of the original square and explain your reasoning.

Of Thanh’s shirts, 60% have short sleeves and the rest have long sleeves. One third of the short-sleeved shirts and one quarter of the long-sleeved shirts are pink. Thanh has 9 pink shirts altogether. How many shirts does Thanh have?

Fill the $3\times4$ grid with numbers from 1, 2, 3, 4, 5 and 6, using each of these numbers at least once. Every row sum must be divisible by 4 and every column sum divisible by 3. What is the greatest possible number of entries equal to 1?

The semifinal of the fifth Titan Mathematics Olympiad has multiple-choice and fill-answer sections. The multiple-choice section contains 10 questions: each correct answer earns 2 points, each wrong answer loses 1 point, and each unanswered question earns 0. Vân scores 12 points in that section and remembers leaving exactly one question unanswered. How many answers did Vân get correct?

An and Bình both travel from $S$ to $F$. An follows $S\to A_1\to A_2\to A_3\to A_4\to A_5\to A_6\to A_7\to A_8\to A_9\to F$, while Bình follows $S\to B_1\to B_2\to F$. Their total distances are equal. Find $A_2A_3+A_4A_5+A_6A_7$, in metres.

A closed cubical box measures $15\times15\times15$ cm. Two small holes are made at the positions shown. What is the greatest volume of water, in $\mathrm{cm}^3$, that the box can hold? Explain your reasoning.

Practice clarification: rest the closed box flat on one of its faces, with both holes unobstructed. Tilting the box is outside this practice model.

Titan plans a mathematics camp for 60 students. Each student registers for 3 classes. Each class has at most 15 students and 1 teacher. Each teacher can teach at most 2 classes. What is the least number of teachers needed?

An writes an equally spaced sequence of five natural numbers. The five remainders when these numbers are divided by 6 are all different. One term is 2024. What is the smallest possible third term?

For 8 weeks of work, Ksi would receive 850,000 đồng and a toy bicycle. Ksi works for 3 weeks and receives 100,000 đồng and the toy bicycle. Find the value of the toy bicycle. Show your reasoning.

Practice clarification: total pay, including the bicycle, is proportional to the number of weeks worked. The bicycle has the same value in both arrangements.