HKIMO 2026 Grade 8 National Preliminary Set 01

A two-digit number $\overline{AB}$ is divisible by 3. Reversing its digits increases it by 63. Find $A+B$.

Jack and Jill leave home together for a pool one mile away. Jill cycles at a constant 10 miles/hour and Jack walks at a constant 4 miles/hour. How many minutes earlier does Jill arrive?

Ice travels 30 km east, then 18 km north, then 6 km west. How far is she from her starting point?

Given $A:B=8:3$, $A:C=6:5$ and $A+B+C=106$, find $B$.

A train travels at 50 km/h. A car travels in the same direction at 80 km/h and takes 45 seconds to pass the train completely. Find the train length in metres.

Editorial convention: neglect the car’s length and treat it as a moving point.

Find $x$ if $(x+1)+(x+2)+\cdots+(x+100)=7450$.

Given $a+b=13$ and $a^2+b^2=75$, find $a^3+b^3$.

How many positive integers $x$ satisfy $27<140-3x$?

Integers $P,G$ satisfy $|P+3G+7|+|2P-3G+5|=0$. Find $P+G$.

The sum of the first n natural numbers, starting at 1, is 210. Find n.

How many three-digit numbers are divisible by 5?

Find the last digit of $2^{2019}$.

Find the number of positive factors of 810.

A rational number $x>0$ satisfies $x=3+\cfrac{4}{3+\cfrac{4}{3+\cfrac{4}{3+\cdots}}}$. Find $x$.

Given $n!=1\times2\times\cdots\times n$, how many trailing zeros does $10!$ have?

What fraction of the square is grey?

Three jars of paint cover all faces of the cube in Figure 1. At least how many jars are needed to cover all faces of the solid in Figure 2?

Editorial convention: read the pictured solid as complete, without additional wholly hidden cubes. Use identical unit cubes and uniform paint coverage, include the underside, and count whole jars.

ABCD is a rectangle and PQRS is a square. The shaded area is half the area of rectangle ABCD. Find PX.

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

A square of side 35 cm is inscribed in a circle as shown. Find the circle area. Take $\pi=22/7$.