Australian AMC 2022 - Senior

The temperature in the mountains was $4^\circ\mathrm C$ but dropped overnight by $7^\circ\mathrm C$. What was the temperature in the morning?

The rectangle shown has area 135. What is the value of $h$ in the diagram?

The value of $5^1+4^2+3^3+2^4+1^5$ is

$\frac1{20}-\frac1{22}=$

Three vertices of a rectangle are the points $(1,4)$, $(7,4)$ and $(1,8)$. At which point do the diagonals of the rectangle cross?

What fraction of this trapezium is shaded?

The sum of the numbers on these six cards is $4\sqrt5$. Lily removes one of them. What is the largest possible sum of the remaining five cards?

In this diagram, what is the value of $x$?

The smaller of these dice has three zeroes and three ones on its faces and the larger has the numbers 1,3,5,7,9 and 11 on its faces. Both dice are rolled once and the numbers showing on top are added. What is the probability of obtaining a sum of 12?

The value of $24^{-1/2}+17^{-1/4}$ is closest to

The operation $\divideontimes$ is defined by $p\divideontimes q=2p-q$. Each of $p$ and $q$ is an integer from $-6$ to $+6$. How many pairs of values $(p,q)$ will have $p\divideontimes q=q\divideontimes p$?

In the diagram, the lengths of the sides of the triangle are 8,9 and 13 centimetres. The centres of the circles are at the vertices of the triangle and the circles just touch. The radius, in centimetres, of the largest circle is

The four integers 3,4,8,11 have their mean and range calculated. A fifth integer is then included that is different from the other four. This doesn’t change the range, but the mean is now an integer. What is this new mean?

Two triangles $ABC$ and $BCD$ are both right angled as shown. Lines $AB$ and $CD$ are parallel. Also $AC=1$ and $AB=2$. What is the length of $CD$?

The value of $x$ in the equation $3^x+3^{x+1}+3^{x+2}=13\sqrt3$ is

Four numbers are equally spaced on the number line, in the given order $\frac1{20},\frac1{22},X,Y$. What is the value of $Y$?

A small equilateral triangle sits inside a larger equilateral triangle as shown. What is the ratio of the areas of the smaller and larger equilateral triangles?

The square pyramid shown is divided into 3 pieces, $P,Q,R$, by two planes that are parallel to the square base. Each of the 3 pieces has the same height. The volume of the piece $P$ is $5\text{ cm}^3$. What is the volume of piece $R$, in $\text{cm}^3$?

A sequence of values $a_1,a_2,\ldots,a_{100}$ is calculated as follows: $a_1=1$, $a_2=2$, $a_3=\frac{a_2+1}{a_1}$, $\ldots$, $a_n=\frac{a_{n-1}+1}{a_{n-2}}$, $\ldots$, $a_{100}=\frac{a_{99}+1}{a_{98}}$. What is $a_{100}$?

A triangular ramp is in the shape of a right-angled tetrahedron. The horizontal base is an equilateral triangle with sides 8 metres. The apex is 1 metre directly above one corner of the base, so that two faces are vertical. In square metres, what is the area of the sloping face?