Australian AMC 2023 - Intermediate

A 40-minute lesson started at 10:50 am. Exactly halfway through the lesson the fire alarm went off. At what time did it go off?

Two rectangles have a vertex in common, as shown. What is the size of the angle marked $x^\circ$ between them?

What is the value of $\dfrac{2+3+4}{7+8+9}$?

How many $25\text{ cm}\times25\text{ cm}$ squares fit in a $50\text{ cm}\times1\text{ m}$ rectangle?

Which one of these is equal to $57\times953$?

A parallelogram $PQRS$ has an area of $60\text{ cm}^2$ and side $PQ$ of length $10\text{ cm}$. Which length is $6\text{ cm}$?

Mei can travel to her grandma’s house by a direct route, or by a scenic route that is 5 km longer. When she travels by the scenic route, and comes directly home, the round trip is 35 km. How long is the direct route?

What is the value of $((2^0)^2)^3$?

What must 0.05 be divided by to get 50?

In the right-angled triangle $ABC$ shown, what is the value of $y$?

The number 11 can be written as the sum of three positive whole numbers in many ways. In how many ways can this be done when the numbers are different and in increasing order?

A two-digit number is reversed and then added to the original number. Which result is impossible?

Amy designed this rectangular flag for her fleet of yachts. What fraction of the flag is shaded?

What is the largest possible whole-number value of $a\times b+\dfrac cd-\dfrac ef$, where $a,b,c,d,e,f$ are the numbers 1, 2, 3, 4, 5 and 6 in some order?

Four children, from youngest to oldest Abdul, Bipin, Cai and Denise, have ages equally spaced apart. Abdul and Bipin’s ages add to 18, while Cai and Denise’s add to 34. How old is Denise?

Exponentia uses six-digit telephone numbers. Only 1000 numbers are currently in use, but the number needed doubles each year. Approximately how many years will it take to run out of phone numbers?

A $15\text{ cm}\times15\text{ cm}$ square of origami paper is dark blue on top and pale yellow underneath. Its top-left corner is folded down along a crease from the top-right corner to a point $x$ cm above the bottom-left corner. After folding, the visible yellow and blue areas are equal. What is $x$?

In the equation $2x+cy=25$, the coefficient $c$ is hidden in the printed diagram, but is a positive integer. The equation has at least one solution with both $x$ and $y$ positive integers. How many different values can $c$ have?

Farmer Smith extended his square property by buying a smaller square of land as shown. The new square increased the total perimeter by 10%. By what percentage did the area increase?

Using the rule that a dog ages 7 dog years per ordinary year, my dog Ruby was born on my ninth birthday. Four years from now, on our birthday, Ruby’s age in dog years will be exactly four times my age in ordinary years. How old am I now?