HKIMO 2026 Grade 9 National Preliminary Set 04

Two trains are 85 m and 90 m long. Travelling in opposite directions, they take seven seconds to pass each other completely. Their speed ratio is 2:3. Find the faster train’s speed in km/h.

Given $A: B=2:3$, $A: C=5:7$ and $A-B+C=288$, find $A$.

How many integers $x$ make $3 x+11$ an even three-digit number?

Each letter in the addition below represents a different digit. Given R = 3, find K.

An archer shoots 30 arrows, earning 10 points per hit and losing 5 points per miss. The total score is 90. How many arrows hit the target?

Find $x+y$ if $|2021-2 x+y|+(x-y-2022)^2=0$.

Find $x$ given that $7=\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$.

How many positive integers $a$ satisfy $a-2<\frac{2 a+20}{a+2}$?

Find $P=2\times 4+4\times 6+6\times 8+\cdots+24\times 26$.

A cubic polynomial $f$ satisfies $f(0)=1$, $f(1)=2$, $f(-1)=2$ and $f(2)=11$. Find $f(-2)$.

Both $n$ and $123!/5^n$ are natural numbers. Find the greatest possible $n$.

Three prime numbers have sum 20. Find their smallest possible product.

Find the remainder when $2023^{111}$ is divided by 9.

Find the last digit of $7^{88}+8^{77}$.

How many positive factors does 2000 have?

Segment AB has endpoints $A=(3+x, y)$ and $B=(-2, x)$ and midpoint $M=(3,6)$. Find $y$.

The figure shows three semicircles and rectangle ABCD. C and D are the centres of the two lower semicircles. Each semicircle has radius 2 cm. Find the shaded area in $\mathrm{cm}^2$.

A triangle has side lengths x,10 and 15. How many integer values can x have?

Find the perimeter of the triangle with vertices $A=(3,4)$, $B=(0,0)$ and $C=(3,-3)$ in rectangular coordinates.

Find an exterior angle of a regular 12-sided polygon.