Question bank AIMO AUSTRALIA 2018

Let $x$ be a single digit. The tens digit in the product of $\overline{2x7}$ and $39$ is $9$. Find $x$.

If $234_{b+1}-234_{b-1}=70_{10}$, what is $234_b$ in base $10$?

The circumcircle of a square $ABCD$ has radius $10$. A semicircle is drawn on $AB$ outside the square. Find the area inside the semicircle but outside the circumcircle.

Find the last non-zero digit of $50!=1\times2\times3\times\cdots\times50$.

Each edge of a cube is marked with its trisection points. Each vertex $v$ is cut off by the plane through the three trisection points closest to $v$. The resulting polyhedron has $24$ vertices. How many diagonals joining pairs of these vertices lie entirely inside the polyhedron?

Let $ABCD$ be a parallelogram. Point $P$ is on $AB$ produced so that $DP$ bisects $BC$ at $N$. Point $Q$ is on $BA$ produced so that $CQ$ bisects $AD$ at $M$. Lines $DP$ and $CQ$ meet at $O$. If $[ABCD]=192$, find $[POQ]$.

Two different positive integers $a,b$ satisfy $a^2-b^2=2018-2a$. What is $a+b$?

Triangle $ABC$ has area $300$. Point $Q$ is the midpoint of $BC$; $P$ lies between $C$ and $A$ with $CP=3PA$; $R$ lies on $AB$ so that $[PQR]=2[RBQ]$. Find $[PQR]$.

Prove that $38$ is the largest even integer that is not the sum of two positive odd composite numbers.

A pair of positive integers is called compatible if one number equals the sum of all digits in the pair and the other equals the product of all digits in the pair. Find all pairs of positive compatible numbers less than $100$ (5 marks).

Investigation (3 bonus marks, separate from the base score): find all pairs of positive compatible numbers less than $1000$ with at least one number greater than $99$.