HKIMO 2020 THPT (Grades 10–12 shared) National Final (Heat) Set 01

$n$ straight lines cut a circle into 361 regions. Find the minimum $n$.

A bag contains $n$ balls of each of five colours: blue, yellow, red, green and purple. After 87 balls are drawn, all remaining balls are red. Find the difference between the largest and smallest possible $n$.

How many integer solutions satisfy $6\lt (5+x^2)/8\lt 9$?

There are 234 balls. Peter and Mary alternately take 5 to 12 balls. Whoever takes the last ball wins; Peter goes first. What is the least number of balls he must add before play to guarantee taking the last ball?

A box contains 30 red socks, 38 white socks and 25 blue socks. How many socks must be drawn to guarantee three pairs that are not white and one additional pair that is not blue?

Find the coefficient of $x^2$ in $(x^2+3x-2)^4$.

Factorize $x^4+x^3-13x^2-x+12$.

Given the repeating continued fraction $\displaystyle x=3-\frac{24}{3+\frac2{3-\frac{24}{3+\cdots}}}$, find its minimum value.

Solve $\log_3(x^{\log_3 36})-\log_3(4^{\log_3 x})=8$.

Find the least positive integer $k$ for which $(3k+2)x^2+(k-2)x-1=0$ has real roots.

Find the smallest positive integer $x$ satisfying $3x\equiv4\pmod5$ and $7x\equiv6\pmod{13}$.

Find the remainder when $37^{30}$ is divided by 17.

Find the last two digits of $A=4+9+14+\cdots+(4+5\cdot98)+(4+5\cdot99)$.

Given $x\gt 0$ and $x=32\sqrt{4\sqrt{32\sqrt{4\sqrt{\cdots}}}}$, find $x$.

If $k^2+(n-782)k-3135=0$ and $n=1+3+5+\cdots+k$, find $k$.

In right triangle $ABC$, $\angle B=90^\circ$ and $\angle C=15^\circ$. Point $D$ is on $BC$ and $AD=CD$. Given $\cos15^\circ=a\sqrt b+c\sqrt d$ with $a,b,c,d\gt 0$, find $a+b+c+d$.

$x$ is acute and $\cos x=24/25$. Find $(\sin x-\tan x)/\sin^2x$.

An interior angle of a regular $n$-gon is $(15n+15)^\circ$. Find the maximum $n$.

A triangle has side lengths 24, 12 and 22. Its area is $\sqrt{a\times b\times c\times d}$, where $a\ne b\ne c\ne d\ne1$. Find $a+b+c+d$.

Find the area enclosed by the $x$-axis and the lines $x=5y-8$ and $y=x+4$.