VTMO 2026 - Grade 6 - Semifinal - SET 10 - Self-study

Tom tells the truth only on dates whose day of the month is divisible by 3 (for example, 3 January, 6 February, …); on all other dates he lies. Jerry tells the truth only on dates whose day of the month is divisible by 4 (for example, 4 January, 8 February, …); on all other dates he lies. In January 2021, on how many dates can both say, “I lied yesterday”?

Define $n!$ (n factorial) as the product of all natural numbers from 1 to $n$. For example, $2021!=1\times2\times3\times\cdots\times2021$. Find the smallest $n$ for which $n!$ is divisible by both 2020 and 2021.

Let $N$ be the product of the odd numbers from 1 to 99, omitting those ending in 5 (5, 15, …): $N=1\times3\times7\times9\times11\times13\times17\times19\times\cdots\times91\times93\times97\times99$. What is the units digit of $N$?

Two football teams, each with 10 players, play each other. One team consists entirely of truth-tellers and the other entirely of liars. Afterwards every player is asked, “How many goals did you score?” Some answer “one”, Châu answers “two”, some answer “three”, and the rest answer “five”. The truth-telling team wins 20–17. Is Châu telling the truth or lying? Explain your reasoning.

In the figure, a circle has diameter 22 cm. A large square has all four vertices on the circle and is divided into eight congruent small rectangles. The four vertices of the smaller square, drawn with dark sides, are the midpoints of the large square’s sides. Find the perimeter of the smaller square.

Calculate $1\times2+3-4\times(5-6)\times7\times8\times9$.

Which of the following numbers is divisible by 6?

There are 30 married couples (60 people altogether) at a party. Each person shakes hands with every other person except their own spouse. How many handshakes take place?

The square in the figure has diagonal 8 cm. What is the area of the shaded region, in square centimetres?

A sequence begins with 1, 2, 3. The fourth term is the sum of the preceding three terms: $1+2+3=6$. Every later term is also the sum of the three terms immediately before it. What is the eighth term?

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

A display shelf initially holds some new products. At the end of day 1, 50% of the products are sold and then 20 are added. At the end of day 2, 50% of the products then on the shelf are sold and then 40 are added. At the end of day 3, 50% of the products then on the shelf are sold and then 80 are added. The shelf now holds the same number of products as initially. How many products were on the shelf initially?

A father is now twice as old as his child. Fifteen years ago, he was three times as old as the child. How old is the child now?

Which pair has greatest common divisor 20?

In the multiplication $\overline{WE}\times\overline{WIN}=\overline{VTMO}$ shown below, different letters represent different digits from 1, 2, 3, 4, 5, 6, 7, 8. Given $V=5$ and that $O,W,E$ are three consecutive natural numbers, find $V+T+M+O$.

A $2\times2\times2$ cube is coloured black and white as shown. Which of the five labelled nets can fold to produce the illustrated cube?

Trọng drives a motorboat upstream from $A$ towards $C$. The boat’s speed is 22 km/h. At $D$ he drops a life jacket, but notices only on reaching $C$, immediately turns back, and retrieves it at $B$. Exactly 1 hour elapses between dropping and retrieving the jacket. The current flows at 2 km/h. Find $BD/CD$.

Model clarification: 22 km/h is the boat’s speed in still water; the jacket drifts with the current and the boat maintains that speed relative to the water.

Two rectangles measuring $8\text{ cm}\times10\text{ cm}$ and $9\text{ cm}\times12\text{ cm}$ overlap as shown. The solid dark region has area 37 cm². Find the area of the striped region.

This semester Minh took five mathematics tests. The mean score of the first two tests was 88, and the mean score of the last three was 93. What was the mean score of all five tests?

The boundary of the figure is made from four semicircles, each of radius 3 cm, arranged as shown. Find the area of the figure. Take $\pi=\frac{22}{7}$.