VTMO 2026 - Grade 6 - Final - SET 1 - Self-study

Twenty small circles are arranged in a ring. Place every element of $S=\{1,2,3,\ldots,20\}$ in exactly one circle, using each number once. Let $A$ be the greatest sum of two neighbouring numbers, $B$ the least such sum, and $C=A-B$. Find the smallest possible $C$ and give an arrangement attaining it.

What is the greatest number of integers that can be selected from $1,2,3,\ldots,161,162$ so that no selected number is twice another selected number? Justify your answer.

Twelve students visit Nam Cát Tiên forest on six tandem bicycles. If two students $A,B$ do not know each other, the number of acquaintances of $A$ plus the number of acquaintances of $B$ is at least $11$.

(a) Prove that at least one pair of students know each other. (b) Prove that the students can be paired so that the two riders on every bicycle know each other.

Clarification: acquaintances are counted among these twelve students, knowing someone is mutual, and a common acquaintance is counted once for each of $A$ and $B$.

Let $S$ be the set of three-digit numbers for which every digit is a prime divisor of the number. Prove that every member of $S$ is divisible by $3$.

In each cell of a $24\times24$ square board, write one of $-24,0,24$. Calculate the sum in each row, each column and each of the two main diagonals. Prove that two of these sums are equal.

An and Bình each travel $600$ m, starting and finishing at the same times. An runs for the first half of the total time and walks for the second half. Bình runs the first half of the distance and walks the second half. Both walk at $5$ km/h, and An runs at $10$ km/h. How fast does Bình run?

At the start of a course, the ratio of boys to girls in Grade 6 at the TITAN mathematics centre is $4:3$. During the first semester, some girls join and the ratio becomes $3:5$. At the start of the second semester, some boys join and the ratio becomes $8:7$. The number of new boys is two less than the number of new girls. No student leaves. How many students are now in Grade 6?

Let $A=2024^{25}+2^{25}+6^{25}$. (a) Is $A$ a perfect square? Explain. (b) Find the remainder when $A$ is divided by $25$.

How many three-digit numbers have three different digits such that the sum of any two digits is a multiple of the remaining digit? Give a complete justification.

The $3\times4$ board is partially filled as shown. Remove one number from $0,2,3,4,5,7,8,10,11$, then use every remaining number once to fill the eight empty cells so that the arithmetic means of every row and every column are all equal. Which number must be removed? Explain and give a valid completed board.

(a) When 14, 53 and 79 are divided by the same positive integer $N$, they leave the same remainder. Find the greatest possible $N$.

(b) The four-digit number $\overline{abab}$ has exactly six positive divisors. Find its smallest possible value.

An, Bình and Châu work together on a landscape painting for their art homework. Working alone, they would finish it in 9, 10 and 12 hours respectively. An and Bình first paint together for 3 hours, then An stops. Bình and Châu finish the painting. How much time does the whole job take?

Model clarification: each person works at a constant rate, and their rates add when they work together.

A batch of fresh fruit is 65% water by mass. After drying for 19 hours, it is 25% water and weighs 15.2 kg less than at the start. How many additional minutes of drying are needed for it to be 5% water? The mass of water removed each minute is constant.

Model clarification: drying removes only water; the mass of dry fruit matter stays unchanged.