HKIMO 2017 Grade 8 National Final (Heat) Set 04

B and C are nonzero digits. The three-digit number $\overline{BCB}$ is divisible by 11, and $\overline{BBC}$ is divisible by 6. Find $\overline{CBC}$.

Andy travels 72 km northwest, 60 km northeast and 27 km southeast. How far is he from his starting point?

Given $\overline{abcd}+\overline{badc}=15532$, find $a+b-c-d$.

It is 12 o’clock. How long until the minute and hour hands overlap again? Give the duration in minutes.

There are 13 white, 11 yellow and 14 brown chopsticks mixed together. What is the least number to draw without looking to guarantee two non-yellow pairs?

Editorial convention: each pair contains two chopsticks of the same colour.

Find $y$ if $|2x+6y+8|+|x-3y+4|=0$.

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

How many negative integers $x$ satisfy $-97\le7x+3\le32$?

Given $a>3$ and real roots of $ax^2+7x-c=0$, find the least possible integer c.

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

Find the remainder when $902^{209}$ is divided by 5.

A positive integer x satisfies $x^2\equiv4\pmod{11}$ and $x>25$. Find the smallest x.

Find the units digit of $A=1+7+7^2+7^3+\cdots+7^{887}+7^{888}$.

How many positive integers less than 6006 are coprime to 6006?

Find the number of positive divisors of 37500.

Right triangles satisfy $\triangle ABC\sim\triangle FED$, with legs $AB=5$ and $BC=12$. Find $DF:EF$.

Find the area of the triangle with vertices $A=(9,4)$, $B=(1,3)$ and $C=(6,8)$.

A cylinder is formed from two circles and a square and has volume $6776\ \mathrm{cm}^3$. Find its total surface area, taking $\pi=22/7$.

An exterior angle of a regular n-sided polygon is $(2n+16)^\circ$. Find n.

A k-sided polygon has 90 diagonals. Find k.