Australian AMC 2009 - Intermediate

$(2000+9)+(2000-9)$ equals

In the diagram, $x$ equals

The value of $\frac12\times\frac23\times\frac34\times\frac45$ is

Which of the following has the largest value?

Which of the following values can replace the box so that

If $3^k=9^{30}$ then $k$ equals

$(x-y)-2(y-z)+3(z-x)$ equals

The point $(k,17)$ lies on the line joining the points $(1,5)$ and $(4,11)$. The value of $k$ is

Paperback books cost \$5 each and hardcover books cost \$7 each. I spend exactly \$86 on books of these two types. What is the maximum number of books that I could have bought?

The figure $PQRS$ is a rectangle divided into 10 squares as shown. The perimeter of this rectangle is 21 centimetres. In centimetres, what is the perimeter of each square?

In a school of 1000 students, 570 are girls. One-quarter of the students travel to school by bus and 313 boys do not go by bus. How many girls travel to school by bus?

In the diagram, triangles $PQT,QTS$ and $QRS$ are isosceles and $\angle PQR$ is a right angle. Angles $PQT$ and $RQS$ are $2x^\circ$ and angle $QTS$ is $5x^\circ$. The value of $x$ is

This pattern consists of squares. Areas, in square units, of four of the squares are shown. Given that $X$ and $Y$ are also squares, their areas are respectively

Two numbers $P$ and $Q$ are such that $P$ is 40% greater than $Q$. The ratio $P:Q$ is

What is the last digit of $6\times8^{2009}$?

Two dice are each numbered from 1 to 6, but are biased so that each is twice as likely to land on any of the even numbers as on any of the odd numbers. The two dice are rolled and the numbers multiplied together. What is the probability that the resulting product is odd?

An eyebrow is an arrangement of the numbers 1, 2, 3, 4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, $(1,3,2,5,4)$ is an eyebrow and $(1,3,4,5,2)$ is not.

The number of eyebrows is

$PQRS$ is a rectangle. $X$ is halfway along $PQ$, $Y$ is a third of the way along $QR$ and $Z$ is a quarter of the way along $RS$. What fraction of the area of $PQRS$ is represented by the quadrilateral $XYZS$?

The difference between a positive fraction and its reciprocal is $\frac9{20}$. The sum of the fraction and its reciprocal is

A circular bottle with dimensions, in centimetres, is shown partially filled with water (figure 1).



The bottle is sealed and then turned upside down (figure 2). The height $h$, in centimetres, of the air in the upturned bottle is