HKIMO 2019 THPT (Grades 10–12 shared) National Final (Heat) Set 02

A bag contains n balls of each of five colours: blue, yellow, red, green and purple. After 67 balls have been drawn, all remaining balls are red. Find the difference between the largest and smallest possible n.

n straight lines cut a circle into 154 regions. Find the smallest possible n.

How many integers x satisfy $-7<(6-x^2)/9<5$?

There are 243 balls. Peter and Mary alternately take 6 to 12 balls, and whoever takes the last ball wins. Peter starts and may add balls before play. What is the least number he must add to guarantee victory?

A box contains 22 red,36 white and 25 blue socks. How many random draws guarantee 3 non-white pairs and 1 non-blue pair?

Convention: a pair uses two socks of the same colour, and the requested pairs do not share socks.

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

If $x=5-\dfrac{18}{2+\dfrac1{5-\dfrac{18}{2+\cdots}}}$, find the minimum value of x.

Editorial interpretation: the repeated fraction is treated as its self-similar fixed-point equation; no particular sequence of finite truncations is specified.

Factorise $x^4+2x^3-9x^2-2x+8$.

Find x if $\log_3(x^{\log_3 12})-\log_3(4^{\log_3 x})=81$.

Find the least positive integer k for which $(k+1)x^2+(k-2)x-11=0$ has real roots.

Find the least positive integer x satisfying $3x\equiv6\pmod5$, $7x\equiv4\pmod{12}$.

Find the remainder when $37^{30}$ is divided by 11.

Find the last two digits of $A=6+13+20+\cdots+(6+7\times98)+(6+7\times99)$.

If $100!/6^n$ is a positive integer, find the largest integer n.

If $k^2+(n-223)k-899=0$ and $n=1+3+5+\cdots+k$, find k.

Find the area enclosed by the x-axis and the lines $x=7y-6$ and $3y=5+2x$.

For acute x with $\sin x=24/25$, find $(1/\cos x-\tan x)\cos^2x$.

An interior angle of a regular n-gon is $(12n+6)^\circ$. Find n.

A 16 cm by 15 cm by 17 cm rectangular container initially has water 2 cm deep. Two cubes of side 7 cm are put inside. What is the maximum rise in water level, in cm?

Model convention: solid, non-floating, axis-aligned cubes; conserved water, no trapped air or water. The question asks for the rise, not the final depth.

A triangle has sides 17,13,12 and area $12\sqrt{a\times b\times c}$, where $a\ne b\ne c\ne1$. Find a+b+c.