TIMO 2018 - Senior Secondary (Grades 10-12) - Preliminary Round - Paper 4

Nonzero digits $A,B,C$ satisfy: $ACB$ is divisible by 9 and $BCB$ is a perfect cube. Find the three-digit number $AAB$.

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

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

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

15 white, 16 yellow and 12 brown chopsticks are mixed together. How many chopsticks must be drawn blindly to guarantee one nonwhite pair and one nonyellow pair?

Compute $9\binom81+9^2\binom82+9^3\binom83+\cdots+9^7\binom87+9^8\binom88$.

Find the coefficient of $x^2$ in $\left(x+\frac1x\right)^4\left(x-\frac1x\right)^5$.

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

Find the smallest integer $a$ such that $(a+2)x^2+ax-7=0$ has no real root.

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

Find the smallest 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 largest integer $n$ for which $\frac{2018!}{20^n}$ is a positive integer.

Find the smallest natural number $n$ satisfying $n^{210}\ge11^{280}$.

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

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

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

Given $\sin x=\sqrt3\cos x$, find $\frac{\tan x}{\cos x}$.

A cuboid container is 18 cm long, 15 cm wide and 12 cm high. Its water is 7 cm deep. A cube of side 9 cm is placed inside. By how many cm does the water rise?