VTMO 2026 - Grade 7 - Semifinal - SET 9 - Self-study

Two vehicles $A,B$ leave Ho Chi Minh City for Vung Tau at the same time, each resting once at a stop 50 km from Ho Chi Minh City. Their total driving times, excluding rest, are 3 hours and 3 hours 20 minutes respectively. When vehicle $A$ has rested for 8 minutes, vehicle $B$ reaches the stop. Find the distance from Ho Chi Minh City to Vung Tau, in km.

A number consists of 2014 digits, all equal to 7. What is the smallest number of digits that must be deleted to obtain a number divisible by 63?

A rectangular prism has base dimensions $3b$ and $6b$ and height $8b$. If its height is reduced by $b$ units, its total surface area decreases by $b\%$ of the original total surface area. Find $b$.

Let $MAB$ be an isosceles triangle with $MA=MB$ and $\angle AMB<90^\circ$. The line perpendicular to $MB$ at $B$ meets ray $MA$ at $N$. Take $C$ on segment $MN$ such that $NC=NB$.

a) Prove that $\angle ABC=45^\circ$.

b) The line perpendicular to $AM$ at $M$ meets ray $BC$ at $D$. Prove that $AD=\sqrt2\,MD$.

Let $P$ be inside the right angle $xOy$. Take $A$ on ray $Ox$ and $B$ on ray $Oy$ such that $\angle APB=90^\circ$. On the rays opposite to $AP$ and $BP$, take $C,D$ respectively so that $CA=OA$ and $BD=OB$.

a) Prove that $C,O,D$ are collinear.

b) Let $M$ be the midpoint of $CD$. Prove that $\angle AMB=90^\circ$.

Chau lies on Friday, Saturday and Sunday and tells the truth on the other days. Minh tells the truth on Tuesday, Wednesday and Thursday and lies on the other days. On which day can both say, “Tomorrow I will lie”?

A train travels from $A$ to $B$ at a constant planned speed. After 70 km, the driver notices an engine fault and proposes reducing the speed by 25%. The captain calculates that this would cause a delay of 1 hour 20 minutes. Instead, he asks the driver to continue at the original speed for another 40 km before making that reduction. The train then reaches $B$ exactly 1 hour late. Find its planned speed, in km/h.

A square has the same perimeter as the right triangle shown, whose perpendicular sides are 8 cm and 6 cm. Find the area of the square.

Find the smallest positive integer divisible by 99 whose decimal representation contains no digit 9.

The diagram is a net of a cube with faces labelled $1,5,6,x,y,z$. The number on each of the faces labelled 5 and $x$ equals the arithmetic mean of the numbers on its four adjacent faces. Find $x$.

Given $P(x)=x^3-4x^2-3x+5$, find $P(2)$.

A football team plays 90 matches in a season. It won 40 of the first 50 matches. How many of its remaining matches must it win to finish with exactly 70% of all matches won?

Vinh takes 75 minutes to travel from $A$ to $B$, while Quang takes 60 minutes. Vinh leaves 9 minutes before Quang. When Quang catches Vinh, they are still 12 km from $B$. How long is $AB$, in km?

Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. Which of the following can equal $3x$?

The natural numbers $a,b$ leave remainders 1 and 2 respectively when divided by 8. What remainder does $9a+3b$ leave when divided by 8?

Car licence plates are numbered from 0001 to 9999. The plate 3681 has the property $3+6=8+1$. How many plates have this property, meaning that the sum of the two left digits equals the sum of the two right digits?

In the diagram, $x=55^\circ$, $y=40^\circ$ and $z=35^\circ$. Find $t$.

In quadrilateral $ABCD$ shown, $\angle ABC=75^\circ$, $\angle BCA=30^\circ$, $\angle CAD=50^\circ$ and $BC=AD$. Find $\angle ADC$.

Find the units digit of $1+4+4^2+4^3+\cdots+4^{100}$.

A teacher distributes 70 candies among 6 students, each receiving at least one, with no two receiving the same number. What is the smallest possible number received by the student with the most candies?