VTMO 2026 - Grade 7 - Semifinal - SET 2 - Self-study

How many rectangles have natural-number side lengths, perimeter $50$, and an area that is a perfect square?

How many three-digit natural numbers have the property that any two adjacent digits differ by $1$? Examples include $123$, $434$ and $987$.

Triangle $ABC$ has $AB=4$, $BC=6$ and $AC=5$. For an arbitrary point $P$, find the minimum possible value of $PA+2PB+3PC$.

In a competition, $25\%$ of all contestants win a prize. Of the male contestants, $\frac27$ win a prize and $420$ do not. There are $648$ female contestants. How many female contestants win a prize?

The positive perfect squares are written consecutively to form $14916253649\ldots$. What is the $100$th digit?

Let $A=0.1\overline{2453}=0.1245324532453\ldots$ be a recurring decimal. Find the least positive integer $k$ such that $kA$ is an integer.

When $1001$ is divided by an odd prime, the remainder is $1$. What is the remainder when $2012$ is divided by that prime?

Replace different letters by different digits so that the sum $\overline{VIETNAM}+\overline{TITAN}+\overline{MATH}$ is as small as possible. The leading digits $V,T,M$ are nonzero. Find $T+I+T+A+N$.

How many prime numbers $p$ have the property that $p^2+59$ has exactly six natural-number divisors? Give a complete justification.

A driver travels from Sài Gòn to Vũng Tàu. The Hoa Mua transport company requires an average speed of at least $48$ km/h over the whole route. Because of poor roads and traffic, the driver averages $40$ km/h over the first half of the distance. What is the least average speed needed over the second half?

Two distinct natural numbers $x,y$ have greatest common divisor $6$ and satisfy $x:y=6:7$. Find $x+y$.

Two positive integers have sum $1323$ and greatest common divisor $147$. How many pairs of such integers are there? Count a pair only once, regardless of order.

An, Bình, Châu and Dũng celebrate Dũng's birthday. The first three guess Dũng's age.

An says: “Dũng is not $10$. He is $11$.”

Bình says: “Dũng is not $11$, and he is not $9$.”

Châu says: “Dũng is not $12$. He is $10$.”

One person makes two true statements, one makes exactly one true statement, and one makes two false statements. How old is Dũng?

For natural numbers $a,b,c,d$, find $a+b+c+d$ given $2^{12}=4^a=8^b=16^c=64^d$.

The average age of a group of people is $15$ years. Three years later, the sum of their ages will be $198$ years. How many people are in the group?

Equilateral triangles $ABC$ and $DCE$ are arranged as shown, with $B,C,E$ collinear in that order. Points $M,N$ are the midpoints of $AC,DE$, respectively. If the area of triangle $ABC$ is $36$ cm², find the area of triangle $BMN$ in cm².

Five distinct lines in a plane are pairwise nonparallel. Exactly one set of three of these lines is concurrent; there is no other triple concurrency. Into how many regions do the five lines divide the plane?

Châu writes a four-digit number and asks Dũng to guess it.

Dũng: “Is it $2896$?” Châu: “One digit is correct and is in the correct position.”

Dũng: “Is it $5273$?” Châu: “Two digits are correct, but both are in the wrong positions.”

Dũng: “Is it $1702$?” Châu: “All four digits are correct, but all are in the wrong positions.”

What number did Châu write?

A runner and a cyclist start together from point $A$ and move clockwise around a circular lake. The cyclist travels at $3.5$ times the runner’s speed. Each time they meet, their location is marked. How many distinct locations are marked?

Natural numbers $a,b$ satisfy $7a+11b=2020$. When $11a+7b$ is as large as possible, find $a\times b$.