VTMO 2026 - Grade 7 - Semifinal - SET 5 - Self-study

Question 1

The figure shows the given angle measures. Find $x$.

1. $40^\circ$
2. $50^\circ$
3. $60^\circ$
4. $70^\circ$
5. $80^\circ$

Question 2

In square $PQRS$, $M$ and $N$ are the midpoints of $PS$ and $RS$, respectively. Triangle $SMN$ has area $18$ cm$^2$. Find the area of triangle $QMN$.

1. $36$ cm$^2$
2. $48$ cm$^2$
3. $54$ cm$^2$
4. $72$ cm$^2$
5. $90$ cm$^2$

Question 3

Let $A$ be the set of natural numbers satisfying both conditions:

The sum of the digits of each number is $18$.

The digits within each number are all distinct.

Let $D$ be the difference between the largest and smallest numbers in $A$. Find the sum of the digits of $D$.

1. $12$
2. $9$
3. $18$
4. $15$
5. Another answer

Question 4

1. $A>B$
2. $A<B$
3. $A=B$
4. $A=-B$
5. $A>-B$

Question 5

Triangle $ABC$ has $\angle A=80^\circ$. The angle bisectors from $B$ and $C$ meet at $I$. Find $\angle BIC$.

1. $80^\circ$
2. $100^\circ$
3. $110^\circ$
4. $120^\circ$
5. $130^\circ$

Question 6

A rectangular prism and a right triangular prism are joined as shown. Find the volume of the combined solid, in cm$^3$.

1. $60$
2. $72$
3. $90$
4. $108$
5. $120$

Question 7

Find $x$ in the figure.

1. $67^\circ$
2. $55^\circ$
3. $59^\circ$
4. $56^\circ$
5. $57^\circ$

Question 8

What is the units digit of $5^5\times7^7\times9^9\times11^{11}$?

1. $0$
2. $1$
3. $3$
4. $5$
5. $7$

Question 9

Let $a,b,c,d,e\in\mathbb N$ satisfy $a^2=b$, $b^2=c$, $c^2=d$, $d^2=e$ and $e^2=2^a$. Find $a$.

Question 10

1. $220$
2. $200$
3. $230$
4. $240$
5. $250$

Question 11

Triangle $PQR$ is right-angled and isosceles at $Q$. The segments $QS,TU,VW$ are perpendicular to $PR$, and $ST,UV$ are perpendicular to $QR$, as shown. What fraction of the area of $PQR$ is shaded?

1. $3/16$
2. $3/8$
3. $5/16$
4. $5/32$
5. $7/32$

Question 12

We have $2024=2^3\cdot11\cdot23$. Find the smallest natural number $A>2024$ that can be written as $A=p^3qr$, where $p,q,r$ are three distinct prime numbers.

Question 13

The letters $a,b,c,d$ represent dates in the calendar shown. Which statement is incorrect?

1. $c-a=d-b$
2. $c=a+7$
3. $d=a+8$
4. $a+c=b+d$
5. $a+d=c+b$

Question 14

The solid shown is made from cubes of volume $1$ cm$^3$ each. What is its volume, in cm$^3$?

1. $20$ cm$^3$
2. $30$ cm$^3$
3. $55$ cm$^3$
4. $25$ cm$^3$
5. $30$ cm$^3$

Question 15

In triangle $ABC$, $\angle A=3\angle B=\tfrac12\angle C$. Find $\angle A$.

1. $30^\circ$
2. $36^\circ$
3. $54^\circ$
4. $60^\circ$
5. $90^\circ$

Question 16

An, Binh and Chau run a $200$ m race, and An finishes first. After $25$ seconds, the total distance covered by the three runners is $500$ m. When An finishes, the total distance Binh and Chau still have to run is $40$ m. How many seconds does An take to finish?

Question 17

Triangle $ABC$ is right-angled at $A$, with $AB=1$ cm and $BC=\sqrt{13}$ cm. Point $M$ is the midpoint of $AC$. Find $\angle BMC$.

1. $120^\circ$
2. $135^\circ$
3. $144^\circ$
4. $150^\circ$
5. $162^\circ$

Question 18

A triangle has side lengths $7$, $9$ and $12$. What kind of triangle is it?

1. Right
2. Equilateral
3. Obtuse
4. Acute
5. Isosceles

Question 19

An and Binh start together from opposite vertices of a square of side $100$ m and walk anticlockwise along its perimeter. An walks at $50$ m/min and rests for $1$ minute whenever reaching a vertex. Binh walks at $40$ m/min and rests for $2$ minutes whenever reaching a vertex. After how many minutes do they first meet?

1. $18$
2. $17$
3. $19$
4. $16$
5. $20$

Question 20

In the five-pointed star shown, the five interior angles of the central pentagon $FGHIJ$ are five consecutive integers in degrees. What is the largest possible value of one of the five tip angles $A,B,C,D,E$ of the star?

1. $37^\circ$
2. $38^\circ$
3. $39^\circ$
4. $40^\circ$
5. $41^\circ$