HKIMO 2026 THPT (Grades 10–12 shared) National Preliminary Set 01

A glass full of water weighs 500 g. When the same glass is half full of water, it weighs 300 g. What is the weight of the empty glass?

17 October 2021 is Sunday. What day of the week was 17 October 2011?

According to the pattern below, find the next number: $1,5,13,29,61,125,\ldots$.

Claudia rolls three six-faced dice simultaneously and writes their sum on the board after each roll. At least how many rolls guarantee that some sum appears more than twice?

Nine people A, B, C, D, E, F, G, H, I sit in a row. B is at one end. H sits adjacent to both F and G. C sits immediately right of D and third to the right of E. A sits immediately left of F. F sits third to the left of B. Who is at the other end?

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

Given $A=2^{2021}$, find $1+2+2^2+2^3+\cdots+2^{2020}$ in terms of $A$.

Let $x,y$ be real. Find $x+y$ when $\sqrt{2x-y}+\sqrt{x+2y-5}$ attains its minimum.

If $\sqrt{17-12\sqrt2}=a-b\sqrt c$ and $a,b,c$ are integers, find the minimum value of $a+b+c$.

How many integer solutions satisfy $x^2\le3x+18$?

Find the number of positive factors of 540.

Find the last digit of $2022^{2021}$.

Given that $8!+9!+10!$ is divisible by $2^n$, find the greatest integer $n$.

Define $x*y=y-x$. If $(a*b)*c=a*(b*c)$, which statement must be true?

How many integers $n$ make $|n^2-2n-3|$ prime?

A solid cuboid is formed by joining 100 cubes of side 1 cm. Find its greatest possible total surface area in $\mathrm{cm}^2$.

How many diagonals does a convex 16-sided polygon have?

Find the minimum distance from $A(3,-6)$ to the line $y=\frac43x$.

In triangle $ABC$, points $M,N$ lie on $BC$ with $BA=BN$ and $CA=CM$. Given $\angle MAN=40^\circ$, find $\angle BAC$.

Two concentric circles have radii in the ratio $1:3$. $AB$ is a diameter of the larger circle. Its chord $BC$ is tangent to the smaller circle, and $AC=6$. Find $BC$.