HKIMO 2018 Grade 8 National Final (Heat) Set 03

According to the pattern, fill the blank: $3,12,27,48,75,108,\_\_,\ldots$.

Peter travels 45 km southeast, 46 km southwest and 22 km northeast. How far is he from his starting point?

If $\overline{abcd}+\overline{dcba}=16775$, find $a+b+c+d$.

Straight lines divide a circular disk into 29 regions. What is the least possible number of lines?

Editorial geometric interpretation: count regions of the disk cut by distinct straight lines, not arcs or curves.

A box has 20 red socks, 7 white socks and 30 blue socks. What is the least number to draw to guarantee two red pairs and two non-red pairs?

Editorial convention: a pair consists of two socks of the same colour.

Find y if $|3x-2y+7|+(x+y-16)^2=0$.

Factorize $x^2-y^2-5x+3y+4$.

How many non-positive integers x satisfy $-108\le15x+9\le65$?

Given $a>5$ and real roots of $ax^2+12x+2c=0$, find the greatest possible integer c.

Given $f(x)=ax^2+bx+c$, $f(-3)=6$, $f(3)=18$ and $f(-1)=2$, find $f(1)$.

Find the remainder when $2019^{513}$ is divided by 5.

A positive integer x has $x^2\equiv2\pmod7$ and $x>31$. Find the smallest x.

Find the units digit of $C=1^{2019}+2^{2019}+3^{2019}+\cdots+512^{2019}+513^{2019}$.

How many positive integers less than 513 are coprime to 513?

Find the sum of all positive divisors of 5544.

Right triangles satisfy $\triangle ABC\sim\triangle DEF$, with legs $AB=18$, $AC=80$. Find $DE/EF$.

Find the area of the triangle with vertices $A=(-5,2)$, $B=(1,6)$ and $C=(4,-2)$.

A cylinder is formed from two circles and a square, with total surface area $8976\ \mathrm{cm}^2$. Find its volume, taking $\pi=22/7$.

An interior angle of a regular n-sided polygon is $(6n+42)^\circ$. Find the largest possible n.

A regular 2019-sided polygon has distance 1 from every vertex to its centre. Find its perimeter, rounded to the nearest integer.