HKIMO 2026 THPT (Grades 10–12 shared) National Preliminary Set 03

Two trains are 85 m and 65 m long. They completely pass each other in 6 seconds when travelling in opposite directions and in 30 seconds when travelling in the same direction. Find the faster speed in km/h.

During Black Friday week, daily sales increase by 20% over the previous day. After two days, by what percentage have sales increased over the original amount?

How many values of $x$ make both $x+2$ and $3x+4$ two-digit primes?

Each letter in the addition below represents a different digit. Find I.

An archer shoots 30 arrows, scoring 10 for an inner-ring hit, 5 for an outer-ring hit and −5 for a miss. His total is 100. What is the minimum possible number of misses?

Find $z$ if $|2020-2x+y|+(x-y-2021)^2+\sqrt{x+y+z}=0$.

For positive real $a$, find the minimum of $\dfrac{4a^2+9}{a+2}$.

How many positive integers $a$ satisfy $a+2<\dfrac{2a+20}{a-2}$?

Find the coefficient of $x^6$ in $(2x^2-1)^{10}$.

If $2x^3+ax^2+bx+3$ is divisible by $2x^2+x-1$, find $a+b$.

Find the largest integer $n$ such that $123!/6^n$ is an integer.

How many three-digit numbers leave remainder 4 on division by 5 and remainder 1 on division by 4?

Find the remainder when $98^{100}$ is divided by 20.

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

Linda concatenates consecutive natural numbers starting at 1: $A=1234567891011121314\ldots$. What is the least number of digits of such a number divisible by 45?

A circle has centre $O(3,4)$ and diameter $MN$, where $M(2+x,y)$ and $N(-2-y,x)$. Find $4x+2y$.

Susan draws a regular $n$-gon and erases two vertices to form an $(n-2)$-gon. The ratio of their interior-angle sums is $10:9$. Find $n$.

The two shortest sides of a right triangle have lengths 6 and 8. Let d and D be its incircle and circumcircle diameters respectively. Find d+D.

Triangle $ABC$ is right-angled at $A$, with $AB=30$, $AC=40$. $M$ is the midpoint of $BC$ and $AH$ is the altitude. Find $MH$.

Figure 1 is made from three identical rectangles of dimensions 3 cm by 1 cm. Cutting along the dotted line produces regions A, B and C in Figure 2. Find the ratio of the area of A to the area of C.