VTMO 2026 - Grade 6 - Semifinal - SET 4 - Self-study

A teacher writes $6$, $80$ and $N$ on three separate slips of paper, where $N$ is even. Tom is asked to find their least common multiple. He mistakes the $6$ for a $9$, but still obtains the correct answer. What is the smallest possible value of $N$?

Rectangle $ABCD$ has perimeter $240\text{ cm}$ and $AB=40\text{ cm}$. Find the area of the shaded part in the diagram.

A digital clock displays the time of day from 00:00 to 23:59. What is the smallest time difference, in minutes, between two displays whose eight digits are all different?

Fill in the blank: $2014\times2015=2014\times2000-2014+2014\times\square$.

Two squares are placed as shown. The larger square has side $8\text{ cm}$, and the striped area exceeds the dotted area by $4\text{ cm}^2$. Find the total area of the unshaded parts, in square centimetres.

On a map, a length of $1\text{ cm}$ represents $8\text{ km}$ in reality. An area on the map is $15\text{ cm}^2$. What is the corresponding actual area, in square kilometres?

The six-digit perfect square $M$ has the form $\overline{aabbcc}$. Given that $M$ is divisible by $5$, find $a\times b$.

All 16 lights in a corridor are initially off. In one operation, a guard changes the state of exactly three lights: an off light is switched on and an on light is switched off. What is the least number of operations needed to turn on all 16 lights?

The bar chart shows a company’s notebook sales by year. In each of 2018 and 2019, sales increased by $60\%$ compared with the previous year. In each of 2020 and 2021, sales decreased by $25\%$ compared with the previous year. Sales in 2021 exceeded those in 2017 by $30800$ notebooks. How many notebooks were sold in 2021?

Three dice each have faces showing 1 to 6 dots. They are arranged as in the diagram. Find the total number of dots on all faces that cannot be seen in this view.

The integers $4,4,x,y,13$ are arranged in nondecreasing order: $4\le4\le x\le y\le13$. How many different pairs $(x,y)$ make the arithmetic mean of the five numbers an integer?

A train travels at $90\text{ km/h}$ and is $300\text{ m}$ long. How long does it take to pass completely through a tunnel $4.5\text{ km}$ long?

A rectangular room measuring $3.75\text{ m}\times2.5\text{ m}$ is paved with equal square tiles. No tile is cut. What is the least number of tiles needed?

Original question: Choose nine different integers from 1 to 36 and place them in the $3\times3$ grid below so that the products in every row, every column and both diagonals are equal. What is the product in each row?

Practice clarification: these conditions do not determine a unique product. Find all possible common products, justify that your list is complete, and give a valid grid for each.

Calculate $(1+1)\left(1+\frac12\right)\left(1+\frac13\right)\left(1+\frac14\right)\cdots\left(1+\frac1{100}\right)$.

Natural numbers $a,b,c$ satisfy $\gcd(a,b)=30$, $\operatorname{lcm}(a,b)=180$, $\gcd(a,b,c)=15$ and $\operatorname{lcm}(a,b,c)=900$. Find the smallest possible value of $a+b+c$.

When asked his age, the mathematician De Morgan (1806–1871) replied, “I was $x$ years old in the year $x^2$.” How old was he in that year?

An and Binh take four tests. Compared with An, Binh scores 1 point more on the first test, 1 point less on the second, and 2 points more on each of the third and fourth tests. By how many points does Binh’s average exceed An’s?

How many four-digit positive integers of the form $\overline{aaaa}$ have exactly eight positive divisors?

A square card is divided into five squares, two of which have equal sides, and one rectangle, as shown. The small shaded square has area $4\text{ cm}^2$. Find the area of the striped rectangle, in square centimetres.