VTMO 2026 - Grade 8 - Semifinal - SET 1 - Self-study

Question 1

Which product below is a perfect square? Recall that $1!=1$, $2!=2\times1=2$, $3!=3\times2\times1=6$ and $4!=4\times3\times2\times1=24$.

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$98!\,99!$
$98!\,100!$
$99!\,100!$
$99!\,101!$
$100!\,101!$

An outer square of area $36\text{ cm}^2$ is divided into four congruent rectangles and one smaller inner square, as shown. The inner square has the same perimeter as one rectangle. Find the area of one rectangle. The figure is not drawn to scale.

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$6\text{ cm}^2$
$6.75\text{ cm}^2$
$7\text{ cm}^2$
$7.5\text{ cm}^2$
$7.75\text{ cm}^2$

Question 3

Simplify $(a-b)(a^2+b^2)(a+b)$.

$a^4+b^4$
$a^3+b^3$
$a^4-b^4$
$a^3-b^3$
$a^4-b^3$

Question 4

Given $x-y=5$, evaluate $x^2(x+1)-y^2(y-1)+xy-3xy(x-y+1)$.

$135$
$153$
$200$
$150$
$180$

Question 5

Given $a^2-b^2=6$ and $(a-2)^2-(b-2)^2=18$, find $a+b$.

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$-2$
$\frac23$
$\frac32$
$2$
$\frac13$

Question 6

Find the smallest prime factor of $100\ldots001$, where exactly twenty zeros lie between the two ones.

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$7$
$11$
$13$
$17$
$19$

Question 7

On a weekday a water park reduces its admission price to attract more visitors. At the end of the day, attendance is $50\%$ higher but total admission revenue is only $20\%$ higher. By what percentage was the ticket price reduced? All tickets have the same price.

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Question 8

Let $x\ne y$ satisfy $x+4y=y^2$ and $y+4x=x^2$. Find $x^2+y^2$.

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Question 9

Three cube-shaped boxes have integer side lengths in centimetres. The first box can contain the second, and the second can contain the third. The first box has volume $218\text{ cm}^3$ greater than the third. Find the volume of the third box, in cubic centimetres.

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Question 10

An, Bình and Châu travel from $A$ to $B$, starting at 6 a.m., 7 a.m. and 8 a.m., with speeds $20$, $30$ and $40$ km/h, respectively. At what time is Châu exactly halfway between An and Bình?

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9 a.m.
9:20 a.m.
10 a.m.
10:20 a.m.
11 a.m.

Question 11

The number $2a+7b$ is divisible by $9$. Find the natural number $n$ such that $5a+nb$ is divisible by $9$.

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Question 12

In acute triangle $ABC$, altitudes $AD$ and $BE$ meet at $H$. Given $DH=2$ cm, $DB=3$ cm and $DC=4$ cm, find the area of triangle $ABC$ in square centimetres.

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$21$
$28$
$35$
$20$
$30$

Question 13

The $4\times4$ square grid shown is made of unit squares. Among pairs of vertices of these unit squares, how many pairs are distance $\sqrt5$ apart?

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$12$
$24$
$36$
$48$
$144$

Question 14

Six different cars, one green, two yellow and three red, are arranged in a row. How many arrangements have no two cars of the same colour next to each other?

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$120$
$60$
$720$
$360$
$36$

Question 15

Find the length of $AB$ in the diagram. The left vertical segment has length $6$ and is perpendicular to $AB$. The successive zigzag segments from its lower endpoint to $B$ have lengths $3,4,5,6,7$; each two consecutive zigzag segments are perpendicular, with their directions as shown.

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$15$
$16$
$17$
$18$
$21$

Question 16

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

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$661$
$665$
$667$
$668$
$669$

$x=12,\ y=8$
$x=10,\ y=6$
$x=11,\ y=8$
$x=12,\ y=4$
$x=14,\ y=16$

Question 20

Let $a,b$ be the two distinct real roots of $P(x)=x^2-10x+2$. Given $a+b=10$, $ab=2$, evaluate $A=\dfrac1{a^3+2a}+\dfrac1{b^3+2b}$.

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$1$
$2$
$\frac12$
$\frac5{12}$
$\frac{12}5$