HKIMO 2018 Grade 7 National Final (Heat) Set 03

Following the pattern, fill in the blank: $2,12,30,56,90,132,\ldots$.

Peter walks 20 km east, then 24 km south, then 13 km west. How many kilometres is he from his starting point?

The four-digit numbers $\overline{abcd}$ and $\overline{badc}$ satisfy $\overline{abcd}+\overline{badc}=15521$. Find $a+b+c+d$.

If $x,y,z$ are prime numbers and $2x^y+y^x=2z$, find the smallest possible value of $z$.

A box contains 10 red socks, 6 white socks, and 22 blue socks. What is the least number of socks that must be drawn at random to guarantee two same-colour pairs whose colours are different?

Solve $(37+21x)-(8x-67)=0$.

Positive integers $x,y$ satisfy $7x-2y=151$. Find the smallest possible value of $|x-y|$.

How many integers $x$ satisfy $-56\le5x+14<89$?

Positive real numbers $a,b$ satisfy $a^2-b^2=51$ and $a=3+b$. Find $a+b$.

Factorise $x^2-2x-63$.

Find the sum of all positive divisors of $513$.

Among the integers from $1$ to $1000$ inclusive, how many are divisible by both $3$ and $5$ but not by $9$?

Find the units digit of $5^{2019}+1^{2019}+3^{2019}$.

Given $x>0$ and $x+\frac1x=5$, find $x^4+\frac1{x^4}$.

Square numbers $A>B$ satisfy $A-B=611$. Find the smallest possible value of $A$.

A cube of side length $18$ is cut into cubes of side length $6$. Find the increase in the total surface area of all the cubes.

A wire forms two circles, each of radius $21$. It is reshaped into three identical rectangles with integer side lengths. What is the maximum area of one rectangle? Take $\pi=\frac{22}{7}$.

Unit cubes are arranged in the layer pattern shown in the figure. Find the total volume of the 15th layer.

The longest side of a triangle has length $12$. Find its greatest possible area.

An interior angle of a regular $n$-gon is eight times an exterior angle. Find $n$.