HKIMO 2018 Grade 9 National Final (Heat) Set 04

Let A, B, C be three nonzero even digits. The three-digit numbers satisfy: $\overline{CBA}$ is divisible by 9, $\overline{CAB}$ is not divisible by 4, and $\overline{ABC}$ is not divisible by 8. Find $\overline{BCB}$.

There are n lines in a plane, no two parallel and no three concurrent. They have 66 intersection points. Find n.

Andy travels 22 km southeast, 17 km southwest, 57 km northwest, then 46 km northeast. How far is he from his starting point, in km?

A competition has 21 problems. Each correct answer earns 3 marks, each wrong answer loses 2 marks, and each blank earns 0. What is the minimum number of candidates guaranteeing that three have the same total score?

Amy reads 1 page on day 1, 2 pages on day 2, 3 pages on day 3, and so on, stopping the first time her cumulative total is a multiple of 90. How many pages has she read?

If $x^2+y^2+17=8x-2y$, find xy.

Factorize $x^2-2y^2-xy-x+5y-2$.

Simplify $\sqrt{28-16\sqrt3}$.

If $x^4+mx^3+nx^2-10x+12$ is divisible by $x^2-4x+3$, find mn.

Find the maximum of $-x^2+14x-25$ for real x.

The five-digit number $\overline{1A0B9}$ is divisible by 11. Find its remainder on division by 9.

Find the remainder of $345^{543}$ on division by 11.

Given $13x\equiv322\pmod{19}$, find the minimum of $|x|$ for integer x.

It is February. Which month will it be after $13^{17}$ months?

Real numbers x,y satisfy $x+y=7$ and $x^2+y^2=29$. Find $x^3-2x^2y-2xy^2+y^3$.

Find the area enclosed by the x-axis, y-axis and $y=\frac43x+7$.

Find the maximum of $\sin^2x+\cos x$ for real x.

An interior angle of a regular $(n^2-n)$-gon is 12° more than 13 times an exterior angle. Given n>0, find n.

A triangle has side lengths 9, 10 and 14 as shown. Find its circumradius.

Find the shortest distance from (2,4) to $y=\frac7{24}x-2$.