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

Find the largest integer $n$ such that the least common multiple of $1,2,3,\ldots,2020$ is divisible by $2^n$.

In the cryptarithm $\overline{VIETNAM}+\overline{TITAN}=\overline{MYSTERY}$, different letters represent different decimal digits. Given $E\lt 3$, find $T+I+T+A+N$.

Evaluate $1+\dfrac{1}{1+\dfrac{1}{1+1}}$.

In a competition, 25% of all contestants receive a prize. Of the boys, $\frac27$ receive a prize and 420 do not. There are 648 girls in the competition. How many girls receive a prize?

A tank holds 4000 litres. An begins pumping water into the empty tank at 20 litres per minute. After 20 minutes, a leak begins at 2 litres per minute while pumping continues. How many minutes after pumping starts will the tank be full?

Some students stand in a circle. A teacher gives out sweets clockwise, beginning with any student: the first receives 1 sweet, the second 2, the third 3, and each successive turn receives one more sweet. Each student receives sweets exactly twice. The total given on the second circuit is 225 more than on the first. How many students are there?

In parallelogram $ABCD$, $E$ lies on $AB$ and $F$ lies on $CD$, with $AE=\frac23AB$ and $CF=\frac23CD$. As shown, $H=AC\cap DE$ and $K=AC\cap DB$. The area of $ABCD$ is $600\text{ cm}^2$. Find the area of triangle $FHK$, in $\text{cm}^2$.

The figure shows 16 circles in a square array, with the 12 circles on the outer border coloured black. If 900 circles are arranged in the same way, how many circles lie on the outer border?

Which number is neither the square nor the cube of a natural number?

Which pair of numbers is relatively prime?

Different letters represent different digits. Given $\overline{ABC}+\overline{ABC}+\overline{ABC}=\overline{CCC}$, find $B$.

A palindrome is a natural number that reads the same forwards and backwards. For example, 1221 is a palindrome. How many three-digit palindromes are there?

The six-digit number $\overline{abcdef}$ is divisible by 7. Prove that the six-digit number $\overline{efabcd}$ is also divisible by 7.

A two-digit positive integer is divided by the sum of its digits. What is the greatest possible remainder?

Five boxes contain the letters shown. Letters are removed so that each box contains exactly one letter and the five remaining letters are all different. Which letter remains in box 5?

Fahrenheit temperature $F$ and Celsius temperature $C$ are related by $F=\frac95C+32$. For example, a freezer compartment at $5^\circ\mathrm C$ corresponds to $\frac95\times5+32=41^\circ\mathrm F$. What Celsius temperature corresponds to $77^\circ\mathrm F$?

A box contains some sweets. They can be shared equally among 3, 4, 6 or 7 children with none left over. What is the least possible positive number of sweets?

How many integer divisors does 20 have?

A number consists of 2014 digits, all of them 7. What is the minimum number of digits to delete so that the remaining number is divisible by 63?

Pi thinks of a five-digit positive integer, then deletes one digit to obtain a four-digit number. The sum of the original number and the resulting number is 52713. Find the resulting four-digit number.