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HKIMO 2026 Grade 9 National Preliminary Set 01

Question 1

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?
  1. 6
  2. 9
  3. 8
  4. 10

Question 2

According to the pattern below, how many circles are in the first 1602 symbols, counting from the left?

Question illustration
  1. 800
  2. 801
  3. 799
  4. 400

Question 3

A mathematics contest has 10 questions. Each correct answer earns 1 point, and wrong or blank answers lose no points. What is the least number of contestants needed to ensure that four have the same score?
  1. 30
  2. 34
  3. 33
  4. 27

Question 4

On Halloween, Hannah’s candies are stolen. Her four guests say:
- Cupcake: “Chocolate stole them.”
- Chocolate: “Jelly stole them.”
- Gummy: “I did not steal them.”
- Jelly: “Chocolate is lying.”

Exactly one guest lies. Who stole the candies?
  1. Cupcake
  2. Chocolate
  3. Gummy
  4. Jelly

Question 5

All 93 students take mathematics, English, or both. There are 70 in mathematics and 54 in English. How many take mathematics only?
  1. 70
  2. 39
  3. 31
  4. 23

Question 6

Integers $P,G$ satisfy $|P+3G+7|+|2P-3G+5|=0$. Find $P-G$.
  1. -4
  2. -1
  3. -5
  4. -3

Question 7

Given $x>0$ and $x+1/x=9$, find $x^3+1/x^3$.
  1. 711
  2. 702
  3. 720
  4. 729

Question 8

Simplify $\sqrt{23+4\sqrt{15}}$.
  1. $\sqrt3+2\sqrt5$
  2. $\sqrt3-2\sqrt5$
  3. $\sqrt5+2\sqrt3$
  4. $\sqrt5-2\sqrt3$

Question 9

How many distinct real numbers $x$ satisfy $(x^2-5)^2=16$?
  1. 4
  2. 2
  3. 1
  4. 0

Question 10

Find $x$: $$\frac{2x}{3+\cfrac4{3+\cfrac4{3+\cfrac4{\ddots}}}}+\frac{3x}{2+\cfrac3{2+\cfrac3{2+\cfrac3{\ddots}}}}=3.$$
  1. 1
  2. 2
  3. 3
  4. 4

Question 11

Find the number of positive factors of 810.
  1. 18
  2. 12
  3. 20
  4. 24

Question 12

Find the last digit of $2^{2019}+3^{2019}$.
  1. 8
  2. 7
  3. 1
  4. 5

Question 13

How many three-digit numbers are divisible by 7?
  1. 127
  2. 128
  3. 105
  4. 130

Question 14

Find the last digit of $3^{2019}+4^{2020}+5^{2021}$.
  1. 8
  2. 2
  3. 5
  4. 4

Question 15

Using digits 0, 1, 5, 6 and 8 without repetition, how many four-digit even numbers can be formed?
  1. 24
  2. 18
  3. 38
  4. 60

Question 16

Combine 36 squares of side 1 cm into a rectangle. Find the smallest possible perimeter.
  1. 26 cm
  2. 24 cm
  3. 20 cm
  4. 36 cm

Question 17

Jack travels 80 km northeast, 40 km northwest, then 50 km southwest. How far is he from his starting point?
  1. 10 km
  2. 30 km
  3. 50 km
  4. 70 km

Question 18

In a regular polygon, an interior angle exceeds an exterior angle by $156^\circ$. How many sides does the polygon have?
  1. 30
  2. 20
  3. 32
  4. 22

Question 19

A fish tank has base dimensions 60 cm by 30 cm and water depth 24 cm. Then 60% of a rock of volume $600\ \mathrm{cm}^3$ is submerged in the water. Find the new water depth.
  1. 24.2 cm
  2. 24.5 cm
  3. 24.25 cm
  4. 24.025 cm

Question 20

Rectangle ABCD is inscribed in the semicircle of diameter FE shown below. Given DA = 16 and FD = AE = 9, find the area of ABCD.

Question illustration
  1. 240
  2. 248
  3. 256
  4. 264