Australian AMC 2010 - Intermediate

The value of $\frac{3\times4}{6}$ is

In the diagram, the value of $x$ is

The value of $2010-20.10$ is

If $m=3$ and $n=-\frac35$, then $\frac mn$ equals

In the diagram,



the value of $x$ is

Consider all the integers from 1 to 100 inclusive. What is the difference between the sum of all the even numbers and the sum of all the odd numbers?

Paul’s three children have birthdays in the same week. He bought a circular birthday cake and divided the cake in proportion to their ages as shown. If none of his children is older than 17, what is the sum of their three ages?



**Interpretation note (AI):** The answer below uses ages in completed whole years. The printed question does not explicitly state this convention; arbitrary fractional ages would make the sum non-unique.

**Source annotation note:** Faint handwritten marks in the supplied image are preserved for fidelity. They are not printed givens or an official answer.

What is the remainder when 2010 is divided by 7?

The areas, in square centimetres, of three rectangles are given.



What is the area, in square centimetres, of the shaded rectangle?

If $\frac16=\frac12+\frac1x$, the value of $x$ is

The perimeter of the equilateral triangle $PQR$ is 48 cm.

What is the perimeter, in centimetres, of the parallelogram $PSTU$?



**Source annotation note:** Faint handwritten marks in the supplied image are preserved for fidelity. They are not printed givens or an official answer.

I am going to the shop with \$4.20 to spend on my favourite chocolates, hazelnut truffles at 30c each and orange truffles at 50c each. I do not want to buy more than twice as many of one as the other. Apart from that, I want to buy as many truffles as I can. How many is that?

Which of the following numbers cannot be expressed as the sum of two or more consecutive positive integers?

Three rectangles are lined up horizontally as shown. The lengths of the rectangles are 2 cm, 4 cm and 8 cm respectively. The heights are 1 cm, 2 cm and 4 cm respectively. A straight line is drawn from the top right-hand corner of the largest rectangle to the bottom left-hand corner of the smallest rectangle.



What is the area, in square centimetres, of the shaded region?

**Source annotation note:** Faint handwritten marks in the supplied image are preserved for fidelity. They are not printed givens or an official answer.

The value of $(123456785)\times(123456782)-(123456783)\times(123456784)$ is

A three-digit number has all digits odd. How many such numbers are divisible by three?

Two squares are drawn inside a circle with centre $O$. The larger square touches the circle at each of its corners. The smaller square touches the circle at two of its corners and one of its sides passes through the centre of the circle as shown.

What is the ratio of the area of the larger square to that of the smaller square?



**Source annotation note:** Faint handwritten marks in the supplied image are preserved for fidelity. They are not printed givens or an official answer.

How many integers $n$ from 2 to 10 inclusive have the property that the sum of any consecutive $n$ positive numbers is odd?

The side length of the grid squares in the figure is 1 cm.



What is the area, in square centimetres, of the shaded triangle?

The 5-digit number $a986b$, where $a$ is the first digit and $b$ is the units digit, is divisible by 72. What is the value of $a+b$?