HKIMO 2026 Grade 9 National Preliminary Set 02

In the sequence below, how many triangles are in the first 1682 figures?

A train travels at 80 km/h and a car travels at 40 km/h in the opposite direction. The car takes 30 seconds to pass the train completely. Find the train length in metres.

Editorial modelling note: as intended by the source options, neglect the car’s length.

A quiz has seven questions. A correct answer earns 2 points; wrong or blank answers lose no points. What is the least number of players needed to guarantee three with the same score?

Seven people say:
- A: Yesterday was Wednesday.
- B: Tomorrow will be Tuesday.
- C: Tomorrow will be Wednesday.
- D: The day after tomorrow will be Tuesday.
- E: Today is Tuesday.
- F: Today is neither Monday, Tuesday nor Sunday.
- G: Today is neither Tuesday nor Saturday.

Exactly three statements are true. What day is today?

Given $A:B=5:6$, $B:C=6:7$ and $A+B+C=54$, find $A$.

Find $x$ if $|3x+2y-5|+|y-2x+8|=0$.

Given $x>0$ and $x+1/x=7$, find $x^2+1/x^2$.

Real numbers $x,y$ satisfy $|x-9|+y^2-4y+4=0$. Find $x+2y$.

Find the minimum value of $x^2+6x-16$.

Simplify $\sqrt{8+2\sqrt{15}}$.

How many positive factors does 2020 have?

Find the units digit of $7^{2016}$.

Given $A=20!=1\times2\times\cdots\times19\times20$, how many trailing zeros does $A$ have?

How many three-digit numbers are divisible by 6?

The sum of the first n natural numbers, starting at 1, is 66. Find n.

Triangle ABC is right-angled at A, and BD is the angle bisector at B. Given AD = 4 cm and DC = 5 cm, find AB.

Combine 2015 squares of side 1 cm into a rectangle. Find its minimum perimeter.

Find the sum of the interior angles of a regular 18-sided polygon.

A cuboid fish tank is 90 cm long, 50 cm wide and 75 cm high, with water depth 45 cm. A brick of volume $18\ \mathrm{dm}^3$ is put inside. What is the new water depth in centimetres?

Editorial modelling note: the source answer assumes the object is fully submerged, with no water spilled.

AFEC is a rectangle in the figure. Given AB:BC = 1:1, AG:GD = 3:1, AC = 8 and CE = 12, find the area of triangle GDE.