Australian AMC 2012 - Senior

The expression $2012-2.012$ equals

By what number must 6 be divided to obtain $\frac13$ as a result?

In the diagram, the sizes in degrees of three angles are given. Find the value of x.

The straight line joining the points $(0,2)$ and $(2,8)$ also passes through

What is the number that is halfway between $\frac ab$ and $\frac cb$?

If $3^{16}\times27^{10}=9^x$, then $x$ equals

If $\frac{p}{p-2q}=3$ then $\frac pq$ equals

Two perfume bottles are similar in shape but one has twice the height of the other. Together they contain 270 mL of perfume. How much perfume, in millilitres, is in the smaller bottle?

If $x$ and $y$ are positive integers, how many integers are greater than $xy$ but less than $x(y+1)$?

The architecture of Federation Square in Melbourne is based on frames as shown in which a large triangle is subdivided into 5 identical triangles, each similar to the large triangle.



If the shortest side of one of the smallest triangles is 1 m, how many metres of framing are required to construct the whole shape?

Five distinct integers are arranged in order, with the smallest being 5 and the largest being 16. The mean of the five integers is prime and is also equal to the median. The number of possibilities for the second largest number is

The average of $3^{3/2}$, $3^{5/2}$ and $3^{7/2}$ is

Triangle $PQR$ is right-angled at $R$. The circle with centre $P$ and radius $PR$ cuts $PQ$ at $S$ and the circle with centre $Q$ and radius $QS$ cuts $QR$ at $T$.



If $T$ bisects $QR$, then the ratio $QS:SP$ equals

In a three-game series of basketball with just two teams, Wollongong and Townsville, the probability that Wollongong will win each game is twice the probability that Townsville will win. What is the probability that Townsville will win the series?

The square shown has side 20 cm. A ‘V’ is shaded as shown.



The area shaded, in square centimetres, is

If the number $10^{2012}-2012$ is written out in full, the sum of its digits is

The side lengths of a right-angled triangle are in geometric progression and the shortest side has length 2. What is the length of the hypotenuse?

Jess marks four of eight identical cards with an $X$ and the other four with a $Y$. She shuffles them and puts them face down in a row and challenges Mike to guess the letter on each card. Mike decides to choose $XYXYXYXY$. What is the probability of Mike correctly guessing all eight cards?

The diagonal of a rectangular tile with integer sides is one-third of the length of the diagonal of a row of tiles identical to the original one. If the perimeter of the single tile is 24 cm, how many tiles are in the row?

In the $\triangle PQR$, $\angle R=2\angle P$, $PR=5$ and $QR=4$. The length of $PQ$ is