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

A teacher writes $90,110,X$ on the board and asks for their least common multiple. Tom misreads 90 as 99, but still gets the correct answer. Find the smallest possible positive integer $X$.

Domain clarification: $X$ is a positive integer; the question does not require the three numbers to be distinct.

Class 6A has 18 boys and 23 girls planting trees. A pupil working alone plants one tree. Groups work according to these rules:
• A group of $n$ boys plants $n+\left\lfloor(n+1)/4\right\rfloor$ trees.
• A group of $k$ girls plants $k+\left\lfloor(k+1)/6\right\rfloor$ trees.
• A mixed group of $p$ boys and $q$ girls plants $p+q+\left\lfloor(p+q+2)/5\right\rfloor$ trees, where $p,q>0$.

How should the class combine these methods to plant the greatest possible number of trees? Find the maximum and explain why. The symbol $\lfloor a/b\rfloor$ means the integer part of $a/b$; for example, $\lfloor5/3\rfloor=1$ and $\lfloor10/5\rfloor=2$.

Model clarification: each pupil participates once, either alone or in one group. More than one group of the same type is allowed. The alternative restriction of at most one group of each type is discussed separately in the solution.

Five teams A, B, C, D and E play a single round-robin football tournament: every pair plays exactly once. A win earns 3 points, a draw 1 point for each team, and a loss 0 points. Their final totals are A: 10, B: 7, C: 6, D: 6, E: 0. Prove that exactly one match is drawn and identify the two teams in that match.

The six-digit number $\overline{abcdef}$ is divisible by 7. Prove that the six-digit number $\overline{efabcd}$ is also divisible by 7. Here $a$ and $e$ are nonzero digits, as required for the two numbers to have six digits.

Mother walks around a lake, completing one lap in 12 minutes. Bình roller-skates in the same direction and overtakes her every 18 minutes. If Bình instead skates in the opposite direction, how many minutes elapse between consecutive meetings?

Model clarification: both keep constant speeds, and Bình's skating speed is unchanged when reversing direction.

Place the digits 1, 2, 3, …, 8, 9 into the nine small circles around the large circle, using each digit once. Read any three consecutive digits clockwise to form a three-digit number. There are nine such numbers, one starting at each small circle. Find their sum.

Square $ABCD$ has side length 10 cm, and $O$ is the midpoint of $AC$. Through $B$, draw a line parallel to $AC$ and choose $K$ on it as shown. Let $M=DK\cap AC$, $N=KA\cap BC$, and $H=KO\cap MN$. Find the difference between the areas of quadrilaterals $CMHK$ and $AOHN$, in square centimetres.

Ten balls are numbered 0 through 9. An takes 3 balls, Bình takes 3 of the remaining balls, and Châu takes the last 4. Each person multiplies the numbers on their balls. Their products are An: 0, Bình: 90, Châu: 72. Find the sum of the numbers on An's balls.

Four distinct natural numbers have these properties:
• The first is a perfect square.
• The second is five times the third.
• The third is prime.
• The fourth is the largest of the four.

Their product is 2025. Find the fourth number.

Tom is chasing Jerry, who is initially 100 m ahead. At their initial constant speeds, Tom expects to catch Jerry after 20 seconds. When the gap has decreased to 15 m, Jerry triples his speed. Tom keeps his original speed and catches Jerry 2 seconds later than originally expected. Find Tom's speed, in metres per second.

Use the nine pieces shown (eight L-shaped pieces and one $2\times2$ square) to tile a $6\times6$ square grid. What is the greatest number of grid-aligned squares, of all sizes, that can contain the marked dot?

Find all pairs of positive integers whose least common multiple is 108 and whose greatest common divisor is 6.

Let $A=2\times3+3\times4+4\times5+\cdots+99\times100$ and $B=2^2+3^2+4^2+\cdots+100^2$. Find $B-A$.