HKIMO 2026 Grade 9 National Preliminary Set 03

In the sequence below, how many triangles are in the first 2023 figures?

Henry and Peter start simultaneously from opposite ends of an 800-metre street, run towards one another, and meet after four minutes. Henry’s speed is three fifths of Peter’s. Find Henry’s speed.

A quiz has five questions. A correct answer earns 1 point, a wrong answer loses 1 point, and a blank earns 0. Negative total scores are allowed. What is the least number of players needed to guarantee three with the same score?

9 January 2023 is a Monday. What is the first later year in which 9 January also falls on Monday?

A glass full of water weighs 500 g. When it is five eighths full, it weighs 350 g. What does the empty glass weigh?

Given $x>0$ and $x+1/x=4$, find $x^3+1/x^3$.

How many integer solutions does $|x-4|=5-2 x$ have?

Real numbers $x, y$ satisfy $x^2+y^2+13=4 x+6 y$. Find $(x-y)^{2023}$.

Find the minimum value of $2 x^2+8 x+11$.

Simplify $\sqrt{17-12\sqrt 2}$.

How many positive factors does 300 have?

Find the last digit of $2023^{2023}$.

Given $15!=1\times 2\times\cdots\times 15$ and that $15!/2^n$ is an integer, find the largest possible $n$.

Editorial domain note: n is taken to be an integer exponent, as intended by this number-theory question.

How many three-digit numbers are divisible by both 4 and 6?

A three-digit number leaves remainders 6, 7 and 10 when divided by 8, 9 and 12 respectively. How many possible numbers are there?

Triangle ABC is right-angled at A. BM bisects angle B, with M on AC. Given AB = 18 and AC = 24, find AM.

Combine 80 squares of side 1 cm into a rectangle. Find the smallest possible perimeter.

In a regular polygon, an interior angle exceeds an exterior angle by $156^\circ$. How many sides does it have?

A cubic tank of side 50 cm contains water to depth 12 cm. A cuboid brick measuring 10 cm by 15 cm by 10 cm is placed inside with one face resting on the tank bottom. What is the smallest rise in water level?

In Cartesian coordinates, $A=(2,5)$, $B=(x, y)$ and $C=(-6,-7)$. If C lies on segment AB and $BC=2 AC$, find $x+y$.