HKIMO 2026 THPT (Grades 10–12 shared) National Preliminary Set 05

Billy has some 1-dollar, 2-dollar and 5-dollar coins. The number of 1-dollar coins is 12 times the number of 2-dollar coins. He has 100 dollars in total. Find the number of 5-dollar coins.

The sum of ten consecutive odd numbers is 300. Find the smallest number.

The four March dates $a,b,c,d$ occupy the cells shown and satisfy $a+b+c+d=40$. What weekday is 1 March?

Find the sum of ten consecutive two-digit integers whose first and last terms are perfect squares.

How many integers $x$ make both $x+7$ and $2x+9$ two-digit numbers?

Evaluate $\displaystyle\frac1{2-\frac1{2-\frac1{2-\frac12}}}$.

Solve $2^0x+2^1x+2^2x+\cdots+2^{2021}x=2^{2024}-4$.

Given $3x+9y=7$ and $9x-3y=-4$, find $x-y$.

The coefficient of $x^2$ in $(2x-3)(x^2-bx)$ is 9. Find the coefficient of $x$.

Find the greatest integer strictly smaller than $A=\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{20}}}}}$.

How many three-digit numbers leave remainder 4 on division by 5 and remainder 5 on division by 6?

The ten-digit number $\overline{A20213101B}$ is divisible by 44. Find $A+B$.

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

How many positive divisors does 1400 have?

Find the largest integer $n$ for which $321!/10^n$ is a natural number.

The figure contains a regular hexagon, a regular pentagon, a square and an equilateral triangle. If $\angle1+\angle3-\angle2=x^\circ$, find $x$.

A rectangle has area 528 and positive integer side lengths. Find its smallest perimeter.

The figure is built from 27 unit cubes. Find its total surface area.

$P(2,7)$ lies on $AB$ and $AP:BP=1:1$. Given $A=(-5+x,4-y)$ and $B=(7-y,6-x)$, find $x^2-y^2$.

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