HKIMO 2018 THPT (Grades 10–12 shared) National Final (Heat) Set 03

According to the pattern, fill the blank: $2,6,15,28,55,78,\_\_$.

n straight lines cut a circle into 232 regions. Find the smallest n.

Prime numbers x,y,z satisfy $6x^y+y^x=3z$. Find the smallest z.

A contest has 31 questions. A correct answer earns 2 points; a wrong or blank answer loses 2 points. What is the least number of candidates that guarantees 3 equal scores? (This is the first of two source questions printed “5”.)

A box has 17 red,13 white and 16 blue socks. How many random draws guarantee 2 non-red pairs and 3 non-blue pairs?

Convention: pairs consist of two same-colour socks; all five selected pairs are disjoint.

If $2x^4-19x^3+mx^2-nx+36$ is divisible by $x^2-4x+3$, find m+n.

If $\sqrt[2019]{x}+1/\sqrt[2019]{x}=2$, find $x^{2019}+1/x^{2019}$.

If $3x=2+\dfrac5{4+\dfrac5{4+\dfrac5{4+\cdots}}}$, find x.

For integer k, find the smallest k for which $(k^2-5)x^2+kx-2=0$ has no real roots.

Factorise $2x^4+5x^3-11x^2-20x+12$.

Find the least positive integer x satisfying $4x\equiv3\pmod{11}$ and $6x\equiv2\pmod8$.

It is September. Which month is $9^{2019}$ months later?

For real x, find the maximum of $1/\sqrt{2x^2+32x+199}$.

If $4000!/15^n$ is a positive integer, find the largest integer n.

The ten-digit number $\overline{20190316AB}$ is divisible by 66. Find the largest A×B.

Find the area enclosed by the x-axis and lines $x=2y-3$ and $2x+y=9$.

Find the area of the quadrilateral with vertices A(−3,5), B(5,7), C(6,−2), D(−2,−1).

A wire makes 3 circles of radius 21. The same wire is cut into 6 identical triangles. Find the maximum area of one triangle, using $\pi=22/7$.

Peter travels 12 km southeast,6 km southwest, then $2\sqrt2$ km east. How far is he from his starting point, in km?

A 15 cm by 10 cm by 18 cm tank contains water 3 cm deep. Eight solid cubes of side 5 cm are put inside. Find the maximum rise in water level, in cm.

Model convention: solid non-floating cubes, faces parallel to the tank; no trapped water or air, and all water is conserved.