VTMO 2026 - Grade 8 - Semifinal - SET 2 - Self-study

Question 1

The sums of the entries in all four $2\times2$ squares of the shown $3\times3$ grid are equal. Find $x$.

Question illustration

Question 2

A 500-page book has pages numbered from 1 to 500. How many occurrences of the digit 5 are used in its page numbers?

$100$
$101$
$200$
$201$
$50$

A space diagonal of a cube joins opposite vertices and passes through its centre, as illustrated. Cube $A$ has a space diagonal twice as long as that of cube $B$. How many times the volume of $B$ is the volume of $A$?

Question illustration

$2$
$4$
$8$
$10$
Another answer

Question 4

Triangle $ABC$ is isosceles at $A$. The internal bisector $BD$ of $\angle ABC$ meets $AC$ at $D$. Its altitude $AH$ satisfies $AH=\tfrac12BD$. Find $\angle BAC$.

Question 5

Six points lie on one of two parallel lines, and four points lie on the other, as shown. How many triangles have vertices among these ten points?

Question illustration

$6$
$15$
$36$
$60$
$96$

Question 6

A rhombus has side length $5$ cm, and its diagonals have ratio $2:1$. Find its area.

Question illustration

$40\text{ cm}^2$
$20\text{ cm}^2$
$30\text{ cm}^2$
$25\text{ cm}^2$
$10\text{ cm}^2$

Question 7

The product of two positive integers is $10000$. Neither integer is divisible by 10. Find their positive difference.

$45$
$210$
$609$
$1242$
Another answer

Question 8

Jerry must run two laps of a badminton court. His average speed on the first lap is $200$ m/min. What average speed is needed on the second lap to achieve an overall average speed of $240$ m/min?

Question illustration

$200$ m/min
$240$ m/min
$280$ m/min
$300$ m/min
$320$ m/min

Question 9

Square $ABCD$ has side $12$ cm. Points $E,F$ are the midpoints of $CD,DA$, respectively, and $G=BD\cap EF$. Find the area of quadrilateral $BCEG$.

Question illustration

$36\text{ cm}^2$
$54\text{ cm}^2$
$63\text{ cm}^2$
$72\text{ cm}^2$
$108\text{ cm}^2$

Question 10

How many three-digit positive integers leave remainder 1 upon division by 9?

$102$
$98$
$101$
$99$
$100$

Question 11

The four dates $a,b,c,d$ occupy the adjacent calendar cells shown, with $a,b$ in one week and $c,d$ directly below them. If $a+b+c+d=52$, find $a+b$.

Question illustration

$19$
$25$
$26$
$27$
$32$

Question 12

What is the remainder when $P(x)=x^3+x^2+x+1$ is divided by $x-3$?

Question illustration

$-17$
$-27$
$40$
$17$
$27$

Question 13

In the figure, $\triangle ABC$ is right-angled at $A$ and $\angle ACB=30^\circ$. Triangles $ABE,ACF,BCG$ are equilateral. Given $S_{ABE}+S_{ABC}=12\text{ cm}^2$, find $S_{ACF}+S_{BCG}$.

Question illustration

$28$
$20$
$21$
$25$
$12\sqrt3$

Question 14

A team plays 90 games in a season and wins 40 of its first 50 games. How many of its remaining games must it win to finish with exactly 70% wins?

Question illustration

$20$
$23$
$28$
$30$
$35$

Question 15

Real numbers $a,b$ satisfy $a^2+4ab=28$ and $2ab-b^2=3$. Find $|a-2b|$.

Question illustration

Question 16

How many right triangles can be formed with vertices among those of a regular 22-sided polygon?

Question illustration

$242$
$200$
$132$
$110$
$220$

Question 17

In square $ABCD$, $M$ is the midpoint of $AD$, and $MN$ is perpendicular to $AC$ with $N\in AC$. Find the ratio of the shaded area $S_{MNC}$ to the area of the square.

Question illustration

$1:5$
$1:6$
$7:36$
$3:16$
$7:40$

Question 18

Polynomial $ax^2+5x+c$, with $a\ne0$, leaves remainder 3 when divided by $5x-3$ and remainder $-3$ when divided by $5x-4$. Find $a+c$.

$5$
$16$
$0$
$-16$
$-5$

Question 19

The number $20142014\ldots2014$ has 1500 digits. What is the greatest number of digits that can be deleted so the remaining number has digit sum 2014?

$860$
$868$
$996$
$611$
$375$

Question 20

Solve $2(x-2024)=4-2024x$.

Question illustration

$2$
$-2$
$\frac12$
$-\frac12$
$0$