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HKIMO 2026 THPT (Grades 10–12 shared) National Preliminary Set 02

Question 1

Michael has $n$ balls labelled with consecutive natural numbers $1,2,\ldots,n$. His brother removes one ball without telling him. The labels on the remaining balls sum to 210. Find $n$.
  1. 20
  2. 21
  3. 19
  4. 22

Question 2

1 November 2020 is Sunday. What day of the week will 1 November 2030 be?
  1. Friday
  2. Thursday
  3. Sunday
  4. Monday

Question 3

According to the pattern below, find the next number: $2,9,28,65,126,217,344,\ldots$.
  1. 345
  2. 471
  3. 513
  4. 561

Question 4

A mathematics test has 10 problems. A correct answer earns 2 marks, a blank answer 0, and a wrong answer loses 2. What is the minimum number of candidates that guarantees two with the same score?
  1. 21
  2. 22
  3. 42
  4. 43

Question 5

There are 10 black socks, 20 yellow socks and 30 blue socks mixed together. Drawing blindly, how many socks are needed to guarantee two nonblack pairs? The two socks in each pair must have the same colour.
  1. 4
  2. 50
  3. 14
  4. 15

Question 6

Let $a$ be real and $a^2-5a-1=0$. Compute $a^4-25a^2-10a+5$.
  1. 4
  2. 10
  3. 5
  4. 6

Question 7

Find $1\times2\times3+2\times3\times4+3\times4\times5+\cdots+13\times14\times15$.
  1. 8190
  2. 10920
  3. 14560
  4. 15670

Question 8

For real $x,y$, find $x+y$ when $x^2-12x+3|2x-3y|$ is minimum.
  1. 10
  2. $-36$
  3. 6
  4. 2

Question 9

Simplify $\sqrt{33-12\sqrt6}$.
  1. $\sqrt6-6$
  2. $6-\sqrt6$
  3. $3-2\sqrt6$
  4. $2\sqrt6-3$

Question 10

How many integers $x$ satisfy $54\le x^2-3x\le180$?
  1. 28
  2. 27
  3. 14
  4. 12

Question 11

Find the number of positive factors of 2020.
  1. 10
  2. 8
  3. 12
  4. 4

Question 12

Find the last digit of $1+3^1+3^2+3^3+\cdots+3^{2020}+3^{2021}$.
  1. 3
  2. 4
  3. 5
  4. 6

Question 13

A three-digit number $n$ leaves remainder 5 on division by 17 and remainder 7 on division by 15. How many possible values of $n$ are there?
  1. 3
  2. 4
  3. 5
  4. 6

Question 14

Find the number of digits of $A=\underbrace{111\ldots111}_{2020\text{ digits}}\times\underbrace{888\ldots888}_{2020\text{ digits}}$.
  1. 4041
  2. 4040
  3. 4038
  4. 4039

Question 15

How many primes $p$ make $p+81$ a perfect square?
  1. 0
  2. 1
  3. 2
  4. 3

Question 16

2020 equilateral triangles of side 1 cm are joined to make a quadrilateral. Find the greatest possible perimeter in cm.
  1. 2022
  2. 4042
  3. 4041
  4. 2021

Question 17

One interior angle of a convex $n$-gon is selected. The other $n-1$ interior angles sum to $2020^\circ$. Find the selected angle.
  1. $130^\circ$
  2. $160^\circ$
  3. $140^\circ$
  4. $150^\circ$

Question 18

Find the minimum distance from $A(0,1)$ to the line $y=-x$.
  1. $\frac{\sqrt2}{4}$
  2. $2\sqrt2$
  3. $\sqrt2$
  4. $\frac{\sqrt2}{2}$

Question 19

How many diagonals does a convex 20-sided polygon have?
  1. 360
  2. 190
  3. 340
  4. 170

Question 20

A circle is inscribed in square $ABCD$ as shown. $M,N,P,Q$ are the midpoints of $AB,BC,CD,DA$, respectively. Find the white area if the shaded area is $43\ \mathrm{cm}^2$. Take $\pi=3.14$.

Question illustration
  1. $57\ \mathrm{cm}^2$
  2. $58\ \mathrm{cm}^2$
  3. $59\ \mathrm{cm}^2$
  4. $60\ \mathrm{cm}^2$