Australian AMC 2015 - Senior

What is the value of $21\times 2015$?

If $K=L+R^2$, $L=4$, $K=85$ and $R$ is positive, then $R$ equals

On their school holidays, Xenia and Ngoc worked for a farmer picking fruit. Xenia worked for 5 days and Ngoc for 3 days. The farmer paid them \$1000, which they shared in the same ratio as the days they worked. Ngoc’s share was

An equilateral triangle, a square and a regular pentagon are joined as in the diagram. What is the size of angle $x$?

$3^{-2}-2^{-3}$ equals

Jenna measures three sides of a rectangle and gets a total of 80 cm. Dylan measures three sides of the same rectangle and gets a total of 88 cm. What is the perimeter of the rectangle?

Two ordinary dice are rolled. The two resulting numbers are multiplied together to create a score. The probability of rolling a score that is a multiple of six is

Two congruent equilateral triangles overlap to make a concave hexagon as shown. Each triangle has a vertex on the other’s centre. What fraction of the hexagon’s area is shaded?

The difference between two numbers is 20. When 4 is added to each number the larger is three times the smaller. What is the larger of the two original numbers?

A bar-tailed godwit was recorded by satellite tag in 2007 to have flown 11 500 km in eight days.

On average, approximately how many kilometres per hour is that?

Let $A$ be the set $\{0,1,2\}$. Let $B$ be the set $\{3,6,x\}$, where $x$ is an integer.

I multiply each number in the first set by each number in the second set.

Let $C$ be the set of all the numbers which are the results of these multiplications.

What could $x$ be such that $C$ has exactly 5 distinct elements?

What is the value of this expression?

$$\frac{1}{\frac{1}{\frac 12+\frac 13}+\frac{1}{\frac 14+\frac 15}}$$

The diagram indicates a pattern of paving blocks that are laid diagonally across a rectangular floor measuring $L\text{ cm}\times 200\text{ cm}$. Each block measures $8\text{ cm}\times 4\text{ cm}$, and any two blocks that touch overlap by 5 cm.

What is the value of $L$?

In a class of 25 students, 11 students are fifteen years old and the rest are sixteen years old. There are 15 boys. There are twice as many sixteen-year-old boys as fifteen-year-old boys. What fraction of the class are sixteen-year-old girls?

Peter was cutting a pipe with an outside diameter of 20 cm. When the cut was just through the wall of the pipe, it was 10 cm in length. How thick was the wall of the pipe in centimetres?

Three different dinosaur fossils have been found in the kingdom of Mathemania. Euleraptor is twice as old as Gaussasaurus, though when Gaussasaurus was alive Euleraptor was twice as old as Fermatops. The sum of the ages of the three fossils is 360 million years. How many million years after Fermatops did Gaussasaurus live?

At Gunaroo High School, a two-weekly (10-day) timetable is used, with 5 periods in each day.

Students take seven subjects, English, Maths, Science, Geography, History, PE and Arts, in a rotating sequence in the order given, starting with English on Day 1, period 1. This takes up 49 of the 50 available periods. The one remaining period is an assembly which is held on the first period of one of the days of the second week. Which lesson can never be held during period 1 of one of the other days of the second week?

Positive integers $x$ and $y$ satisfy $(2^x+1)(2^y-1)=2015$. What is the value of $xy$?

Given an integer N greater than 1, the sum of N and the second largest factor of N can be found. For example, with N = 55, the sum is 55 + 11 = 66. For how many integers is this sum equal to 42?

Unequal squares $PQRS$ and $STUV$ are aligned with the straight line $PST$. $XQ$ and $YU$ are perpendicular to $XY$. The length $XQ+YU$ is the same as