Australian AMC 2017 - Intermediate

The remainder when 2017 is divided by 5 is

Eight children in a swimming race are John, Iain, Hans, Ivan, Giovanni, Beth, Liz and Elisa. They are put in lanes 1 to 8 randomly. What is the probability that Beth, Liz or Elisa is in lane 1?

What is the total shaded area of this diagram, in square units?

1000% of 1 is

The diagram shows six angles, with three of them equal to 30°. The remaining three angles in the diagram are all equal to x°. What is the value of x°?

Which of the following fractions is largest?

Alice is playing with words. At each tick of her grandfather’s clock she swaps two letters. What is the smallest number of clock ticks during which she can change WORDS to SWORD?

How many ways are there of placing a single 3 × 1 rectangle on this grid so that it completely covers three grid squares?

Suppose 3 a = 4 and 9 b = 7. Then 18(a + b) is equal to

Mike receives a number of emails each day. One Friday, he notices that for five days in a row, the number of emails he has received is a different prime number over 20. What is the least number of emails he could have received in these five days?

In the triangle below, PQ = SQ = SR = QR. The ratio \(\angle PSR : \angle PQS\) is equal to

A rectangular swimming pool is 50 metres long and 20 metres wide. It is divided into ten lanes, each 50 metres long and 2 metres wide, numbered in order from 1 to 10. I notice that if I swim one lap down the middle of the lane, then walk back around the edges of the pool, I have to walk 20% further than I swim. Which of the following could be my lane number?

A square table top is covered with identical square tiles. A total of 25 tiles are used to form the two diagonals. How many tiles are used on the table top?

Gwen’s boots have 5 pairs of holes for the laces. Opposite holes are 4 cm apart and the first and last hole on each side are 12 cm apart. Gwen laces and ties each boot as shown, where the knot and bow use a total of 40 cm of bootlace. Approximately how long is the lace?

The number \(\sqrt{20-\sqrt{14+\sqrt{5-\sqrt{1}}}}\) is closest to

This square sheet of paper PQRS measures 40 cm × 40 cm. The top-left corner P is folded down to meet M, the midpoint of the bottom side RS, making a single straight crease. Where does the crease cross QR, the right-hand side of the sheet of paper?

The numbers 18, p, 13, q, 15, r, 7 have a mean of 11. The mean of p, q and r is:

The identity \(a(a^9-a^8)+a^9=a^x\) is true for all a. The value of x is

All the digits from 0 to 9 are used exactly once to form two 5-digit numbers. What is the smallest possible difference between these numbers?

A cube of surface area X is sliced into two rectangular prisms. One of the prisms has surface area \(\frac 12 X\). What is the surface area of the other prism?