Australian AMC 2021 - Senior

Each small triangle is the same size. What fraction of the largest triangle is shaded?

When 2021 is divided by 7, what is the remainder?

What is $12^1-12^{-1}-12^0$?

In this diagram, what is the size of each angle marked $\theta$?

What is the value of $\dfrac{(20\times21)+21}{21}$?

Henry’s electric scooter took him 1.5 km in 3 minutes and 45 seconds. What was his average speed in kilometres per hour?

Evaluate $\dfrac{8^3\times3^6}{6^5}$.

The product of the recurring decimals $0.\overline3$ and $0.\overline6$ is $0.\overline x$. What is the digit $x$?

The parallelogram shown has area 48 square units, with $P(1,2)$, $Q(a,2)$ and $S(4,8)$. What is $a$?

Mervin may paint the four walls and ceiling of his rectangular bedroom as he wishes. Each surface is painted in one of three colours, red, white and green, and no two adjacent surfaces may have the same colour. How many different ways can he paint the room?

Here is an ascending list of fractions whose denominators, in simplest form, are less than 6: $\frac15,\square,\frac13,\frac25,\square,\square,\frac23,\frac34,\frac45$. Three fractions have been omitted. What is their sum?

In autumn, Tilly’s meadow changes rapidly with the weather. The number of flowering plants starts at 150 000 and halves each week as plants die off. The number of fungi starts at 20 and triples each week. To the nearest week, how long will it be until the fungi outnumber the flowering plants?

A physics formula is $r=\dfrac{mV}{qB}$. If $q$ is trebled, $m$ is halved, and $r$ and $B$ stay unchanged, what happens to $V$?

In the diagram, $AD=12$ and $ABC$ and $CDE$ are right-angled isosceles triangles. Triangle $BDE$ has area 9. What is the area of triangle $ABD$?

Which lists $1^{40},2^{30},3^{20},4^{10}$ in increasing order?

In rectangle $ABCD$, the marked lengths $AE=x$, $EB=y$ and $AD=z$ are positive integers. Triangle $AED$ has area 12 square units and triangle $BCE$ has area 21 square units. How many possible values of $z$ are there?

A square is divided into three congruent isosceles triangles and a shaded pentagon, as shown. What fraction of the square’s area is shaded?

A non-standard die has the numbers 2, 3, 5, 8, 13 and 21 on its faces. The die is rolled twice and the numbers are added. What is the probability that the sum is also a value on the die?

If $a$ and $b$ are positive, what is $\sqrt{a^2+\frac1{b^2}}\times\sqrt{b^2+\frac1{a^2}}$ equal to?

The quadrilateral shown is cut into two equal areas by the dashed line. Its parallel bases have lengths 4 and 6, and the perpendicular left side is divided into lengths 1 and 1. The right side is divided into upper length $a$ and lower length $b$. What is $a:b$?