HKIMO 2026 THPT (Grades 10–12 shared) National Preliminary Set 04

According to the pattern, fill the blank: $1,2,6,12,25,48,\boxed{?},\ldots$.

A circle is cut by n straight lines into 29 distinct regions. Find the minimum n.

Primes $x,y,z$ satisfy $2x^y+y^x=z$. Find the smallest $z$.

A competition has 20 questions. Each correct answer earns 3 points, each wrong answer loses 1 point, and a blank earns 0. What is the minimum number of contestants guaranteeing three with the same score?

A box holds 15 red, 21 white and 17 blue socks. How many socks must be drawn at random to guarantee two pairs that are not white and two pairs that are not red?

Find the largest value of $x$ given $x=3-\dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-\cdots}}}$.

Real numbers $x,y$ satisfy $x^2+y^2+5=4x+2y$. Find $x-y$.

If $\sqrt{18-12\sqrt2}=a\sqrt3-\sqrt b$ for positive integers $a,b$, find $a+b$.

If $x^4-8x^3+mx^2-nx+8$ is divisible by $x^2-6x+8$, find $m$.

Find the maximum of $-x^2+12x-19$.

The six-digit number $\overline{20A1B8}$ is divisible by 12, with digits $A<B$. Find the smallest $A+B$.

For real $x$, find the maximum of $A=\dfrac{10}{\sqrt{x^2+14x+99}}$.

Given $7x\equiv107\pmod{11}$, find the minimum integer value of $|x|$.

It is May now. Which month will it be after $7^{20}$ months?

Real numbers $x,y$ satisfy $x-y=6$ and $x^2+y^2=40$. Find $2x^3-2y^3$.

Find the area bounded by the $x$-axis, the $y$-axis and $x=9-\frac38y$.

Peter walks 8 km northeast, 4 km northwest, then 6 km southwest. How far is he from his starting point?

An interior angle of a regular $n$-gon is 12 degrees more than six times an exterior angle. Find $n$.

Find the area of triangle $ABC$ with $A(-3,6)$, $B(-1,3)$, $C(1,-5)$.

Find the shortest distance from $(2,5)$ to the line $4y=3x-6$.