HKIMO 2018 Grade 9 National Final (Heat) Set 03

Find the missing number in the pattern $1,2,6,12,25,48,\underline{\quad},\ldots$.

Straight lines divide a circular disk into 154 regions. Find the minimum number n of lines.

Primes x, y, z satisfy $2x^y+y^x=z$. Find the minimum z.

A contest has 20 problems. Each correct answer earns 3 points, each wrong answer loses 1 point, and each blank earns 0. How many candidates are needed to guarantee that 3 have the same total score?

A box contains 15 red socks, 21 white socks and 17 blue socks. What is the least number of blindly drawn socks guaranteeing 2 nonwhite pairs and 2 nonred pairs? Each pair uses two socks of the same colour.

Given the repeating expression $x=3-\dfrac2{3-\dfrac2{3-\dfrac2{3-\cdots}}}$, find the sum of the possible values of x.

Source interpretation unresolved: the printed key 3 is the sum of the formal fixed-point roots 1 and 2. Ordinary finite truncations of the displayed infinite continued fraction converge only to 2. The source does not distinguish these meanings; manual review is required and no unconditional answer is assigned.

Factorize $x^2-y^2+x-7y-12$.

Positive integers a, b satisfy $\sqrt{18-12\sqrt2}=a\sqrt3-\sqrt b$. Find a+b.

If $x^4-8x^3+mx^2-nx+8$ is divisible by $x^2-6x+8$, find m+n.

Find the maximum of $-x^2+12x-19$ for real x.

The six-digit number $\overline{20A1B8}$ is divisible by 12 and A>B. Find the sum of the distinct possible values of A+B.

For real x, find the maximum of $1/\sqrt{x^2+14x+99}$.

An integer x satisfies $7x\equiv107\pmod{11}$. Find the minimum of $|x|$.

It is May now. What month will it be after $7^{20}$ months?

Real x, y satisfy $x-y=6$ and $x^2+y^2=40$. Find $2x^3+3x^2y-3xy^2-2y^3$.

Find the area enclosed by the coordinate axes and $x=9-\frac38y$.

Peter travels 8 km northeast, 4 km northwest and $\sqrt2$ km west. How far is he from his starting point?

An interior angle of a regular n-gon is 12° more than 6 times an exterior angle. Find n>0.

Find the area of the triangle with vertices $A=(-3,6)$, $B=(-1,3)$, $C=(1,-5)$.

Find the shortest distance from (1, 5) to $y=\frac5{12}x-6$.