Australian AMC 2008 - Senior

The value of $8002-2008$ is

The difference between $\frac1{20}$ and $\frac2{10}$ is

In the diagram, $x$ equals

The value of $\dfrac{200\times8}{200\div8}$ is

The smallest value that $x^2-4x+3$ can have is

\$3 is shared between two people. One gets 50 cents more than the other. The ratio of the larger share to the smaller share is

When $1000^{2008}$ is written as a numeral, the number of digits written is

A semicircle is drawn on one side of an equilateral triangle. The ratio of the area of the semicircle to the area of the triangle is

Given that $\cos x=0.5$ and $0^\circ<x<90^\circ$, which of the following has the greatest value?

A fishtank with base 100 cm by 200 cm and depth 100 cm contains water to a depth of 50 cm. A solid metal rectangular prism with dimensions 80 cm by 100 cm by 60 cm is then submerged in the tank with an 80 cm by 100 cm face on the bottom.

The depth of water, in centimetres, above the prism is then

Which of the following numbers is the largest?

A normal die is thrown 100 times. The sum of the numbers obtained will most likely be

What is the smallest whole number which gives a square number when multiplied by 2008?

AI editorial clarification: Zero satisfies the literal whole-number wording but is not listed. For this imported multiple-choice item, interpret the requested multiplier as positive.

A cross is made up of five squares, each with side length 1 unit. Two cuts are made, the first from $X$ to $Y$ and the second from $Z$ to $T$, so that $ZTX$ is a right angle. The three pieces are then arranged to form a rectangle.

What is the ratio of the length to the width of the rectangle?

A function is said to be a toggle function on $(p,q,r)$ if $f(p)=q$, $f(q)=r$ and $f(r)=p$. The function $f(x)=ax^2+bx+c$ is a toggle function on $(1,2,3)$. What is the value of $c$?

Two conical rollers with perpendicular axes touch on a line that is $30^\circ$ to the axis of the smaller roller and $60^\circ$ to the axis of the larger roller. If the larger roller makes 1 revolution per second and there is no slipping, how many revolutions per second does the smaller roller make?

Consider the set $X=\{1,2,3,4,5,6\}$. How many subsets of $X$, with at least one element, do not contain two consecutive integers?

Farmer Taylor of Burra has two tanks. Water from the roof of his farmhouse is collected in a 100 kL tank and water from the roof of his barn is collected in a 25 kL tank. The collecting area of his farmhouse roof is 200 square metres while that of his barn is 80 square metres. Currently, there are 35 kL in the farmhouse tank and 13 kL in the barn tank.

Rain is forecast and he wants to collect as much water as possible. He should:

A sequence $\{u_1,u_2,\ldots,u_n\}$ of real numbers is defined by $u_1=\sqrt2$, $u_2=\pi$, $u_n=u_{n-1}-u_{n-2}$ for $n\ge3$. What is $u_{2008}$?

In the diagram, $RU$ is equal in length to $ST$. What is the ratio of the area of $\triangle QRU$ to the area of $\triangle QST$?