HKIMO 2017 THPT (Grades 10–12 shared) National Final (Heat) Set 04

Let $A,B,C$ be nonzero digits. The three-digit number $\overline{ACB}$ is divisible by $9$, and $\overline{BCB}$ is a perfect cube. Find $\overline{AAB}$.

There are $n$ pairwise nonparallel lines in a plane, with no three concurrent. They have $105$ intersection points. Find $n$.

Andy walks $99$ km north, $15$ km east, $78$ km south and $87$ km west. How far, in km, is he from his starting point?

1 September 2018 was Saturday. What day of the week is 21 September 2078?

Editorial working convention: Use the Gregorian calendar.

There are $15$ white, $16$ yellow and $12$ brown chopsticks mixed together. With eyes closed, how many must you take to guarantee one pair that is not white and another pair that is not yellow?

Editorial working convention: A pair consists of two sticks of the same colour; the two pairs must use four distinct sticks.

Calculate $\binom81+9\binom82+9^2\binom83+\cdots+9^6\binom87+9^7\binom88$.

Find the coefficient of $x$ in $(x+1/x)^4(x-1/x)^5$.

Evaluate $\sqrt{6-\sqrt{6-\sqrt{6-\cdots}}}$.

Editorial working convention: Use the limit of finite nested radicals, starting from an innermost zero.

For integer $a$, find the smallest $a$ such that $(a+2)x^2+ax-7=0$ has no real roots.

Factorise $6x^4-x^3-14x^2-x+6$.

Find the least positive integer $x$ satisfying $3x\equiv1\pmod5$ and $5x\equiv1\pmod7$.

It is June now. What month will it be after $7^{38}$ months?

How many positive integers $x$ make $\sqrt{x^2-22x+221}$ an integer?

Find the greatest integer $n$ for which $2018!/20^n$ is a positive integer.

Find the least positive integer $n$ such that $n^{210}\ge11^{280}$.

Find the area bounded by the $x$-axis and the lines $4x-3y=1$ and $y=-3x/5-2/5$.

A triangle has side lengths $2x,3x,4x$ and incircle radius $r$. Find $x/r$.

In triangle $ABC$, $BC=10$, $AB=13$, and $\angle ACB=45^\circ$. Find its area.

Given $\sin x=\sqrt3\cos x$, find $\left|\tan x/\cos x\right|$.

A rectangular container measures $18$ cm by $15$ cm by $12$ cm high. Water is initially $7$ cm deep. A cube of side $9$ cm is put inside. By how many cm does the water level rise?

Editorial working convention: The solid cube rests flat on the bottom and absorbs no water.