HKIMO 2026 Grade 9 National Preliminary Set 05

Given $A: B=1:2$, $B: C=3:4$, $C: D=5:6$ and $A+B+C+D=1064$, find A.

There are n lines in a plane. No two are parallel and no three pass through the same point. If they have 120 intersection points, find n.

Andy travels 22 km southeast, 18 km southwest, 34 km northwest and 23 km northeast. How far is he from his starting point?

When do the hour hand and minute hand coincide between 9 o’clock and 10 o’clock?

Find the sum of all real solutions of $x^{100}+x^{98}+\cdots+x^4+x^2=2023$.

If $x^2+y^2+17=8 x-2 y$, find xy.

When $ax^3+bx^2+cx+d$ is divided by $x^2+4 x+2$, the quotient is $2 x+1$ and remainder $x-4$. Find $a+b+c+d$.

Simplify $\sqrt{28-16\sqrt 3}$.

Given $a+b=8$, $a^2+b^2=34$ and $a>b$, find $a-b$.

Find the minimum value of $x^2+8 x-16$.

The five-digit number $\overline{1 A 0 B 9}$ is divisible by 11. Find its remainder when divided by 9.

Find the remainder when $2^{2023}$ is divided by 11.

Given $3 x\equiv 537\pmod{13}$, find the smallest three-digit integer x.

It is February now. What month will it be after $13^{17}$ months?

How many four-digit numbers have a digit sum that is a multiple of 9?

Find the area enclosed by the x-axis, y-axis and the line $y=\frac 43 x+8$.

In Cartesian coordinates, $A=(-3,-2)$, $B=(-1,7)$, $C=(1,9)$ and $D=(2,4)$. Find the area of quadrilateral ABCD.

A regular polygon has $n^2-n$ sides. An interior angle is 12 degrees more than 13 times an exterior angle. Given n>0, find n.

A triangle has side lengths 9, 12 and 15. Find the radius of its incircle.

Find the shortest distance from the origin O to the line $3 y=4 x-15$.