VTMO 2026 - Grade 6 - Semifinal - SET 3 - Self-study

A sequence begins with 6. If a term t is even, the next term is $t/2$; if a term s is odd, the next term is $3s+1$. Thus its first four terms are 6, 3, 10, 5. Find the 100th term.

Let $A=\frac12+\frac13+\frac14+\cdots+\frac1{2025}$ and $B=\frac1{2024}+\frac2{2023}+\frac3{2022}+\cdots+\frac{2023}{2}+\frac{2024}{1}$. Write $A/B=a/b$, where a and b are positive integers with greatest common divisor 1. Find a + b.

The integers 4,4,x,y,13 satisfy $4≤4≤x≤y≤13$. How many different pairs (x,y) make the average of these five integers an integer?

Calculate $1+2-3+4+5-6+7+8-9+10+11-12+13+14-15$.

P is the product of the first 2017 prime numbers. What is the units digit of P?

Which of the following calculations has an even result?

How many prime numbers are greater than 50 and less than 100?

Two positive integers have product 10000, and neither is divisible by 10. Find their difference.

Two cars leave simultaneously, one from Ho Chi Minh City towards Vũng Tàu and the other in the opposite direction. They meet at a point one third of the total route from Vũng Tàu. Find the ratio of the speed of the car from Ho Chi Minh City to the speed of the car from Vũng Tàu.

Tap A can fill a tank in 15 minutes, and tap B can fill it in 10 minutes. How long will both taps, opened together, take to fill the tank?

Fill the six circles with 4,5,6,7,8,9, using each number once; 7 is already filled in. The sum of the numbers in every pair of adjacent circles must be prime. Which number goes in the circle marked ?

Twenty students take a test. No two students have the same number of correct answers, and each question is answered correctly by at most three students. What is the smallest possible number of questions?

An, Bình and Châu race 100 m at constant speeds. When An finishes, Bình is 10 m from the finish and Châu is 20 m from it. When Bình finishes, how many metres is Châu from the finish?

A job is assigned to An, Bình and Châu. An alone takes the same time as Bình and Châu working together. Bình alone takes twice as long as Châu and An together. Châu alone takes t times as long as An and Bình together. Find t.

A circle and a quadrilateral are drawn in a plane. What is the greatest number of regions into which they can divide the plane?

A square has side length 14 cm. A circle is inscribed in it, and a smaller square is inscribed in the circle as shown. Find the area of the shaded region.

Let $A=2^6×3^2×5^4$. Find the sum of the digits of A.

Find the units digit of $1+4+4^2+4^3+\cdots+4^{100}$.

An and Bình take four tests. Bình scores 1 point more than An on the first, 1 point less on the second, and 2 points more on each of the third and fourth. How many points higher is Bình’s average than An’s?

A 4 × 4 grid has one lamp in each square. Touching a lamp changes its state and the states of all lamps in squares sharing a side with its square: on becomes off and off becomes on. Initially all lamps are on. What is the fewest lamps that need to be touched to turn all lamps off?