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Australian AMC 2017 - Senior

Question 1

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?

Question illustration
  1. $\frac38$
  2. $\frac12$
  3. $\frac35$
  4. $\frac53$
  5. $\frac13$

Question 2

What is the value of $17a+20b$ when $a=20$ and $b=17$?
  1. 680
  2. 689
  3. 1720
  4. 2017
  5. 3737

Question 5

This pinwheel star is formed by rotating a right-angled triangle around one of its corners. What is the angle at each of the nine tips that are marked with dots? The nine congruent triangles meet at the centre and fill one full turn there.

Question illustration
  1. $30^\circ$
  2. $40^\circ$
  3. $45^\circ$
  4. $50^\circ$
  5. $60^\circ$

Question 6

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?
  1. 3
  2. 4
  3. 6
  4. 7
  5. 8

Question 7

$200-199+198-197+196-\cdots+2-1$ is equal to
  1. 1
  2. 99
  3. 100
  4. 101
  5. 200

Question 8

I type 360 characters per minute. I am typing a long document that has exactly 50 characters per line. After 8 minutes of typing, which line number am I on?
  1. 54
  2. 55
  3. 56
  4. 57
  5. 58

Question 9

I make a multiplication table for the numbers from 1 to 5. What is the sum of the 25 numbers inside this table? Each interior entry is the product of its row and column labels. The source table is blank for these entries.

|$\times$|1|2|3|4|5|
|---|---|---|---|---|---|
|1||||||
|2||||||
|3||||||
|4||||||
|5||||||



Question illustration
  1. 100
  2. 200
  3. 225
  4. 250
  5. 256

Question 10

Evaluate $\sqrt2\times\sqrt[4]2\times\sqrt{800}\times\sqrt[4]8$.
  1. 2
  2. 4
  3. 8
  4. 80
  5. 800

Question 11

Each row, column and diagonal of this $3\times3$ grid should add up to 30. What number goes in the bottom-right corner? The given entries are 2 in the top-middle cell, 8 in the middle-left cell and 12 in the middle-right cell.

||2||
|---|---|---|
|8||12|
|||?|



Question illustration
  1. 3
  2. 15
  3. 5
  4. 13
  5. 4

Question 12

The vertices of the innermost white square bisect the sides of the shaded square. What is the area of the innermost white square? In the diagram, each side of the outer square is divided by a vertex of the shaded square into lengths 1 and 3, in that order when going clockwise around the outer square.

Question illustration
  1. 6
  2. 4
  3. 10
  4. 3
  5. 5

Question 13

Two cars leaving Sale for Melbourne are travelling at $60\text{ km/h}$ and are $240\text{ m}$ apart travelling west on the Princes Highway. When each car reaches the $80\text{ km/h}$ sign it starts travelling at $80\text{ km/h}$. How far apart are they once they are both past the sign?
  1. $180\text{ m}$
  2. $240\text{ m}$
  3. $360\text{ m}$
  4. $300\text{ m}$
  5. $320\text{ m}$

Question 14

In the triangle shown, $PQ=SQ=SR=QR$ and $\angle STR=90^\circ$. The points $P,Q,T,R$ lie on one straight line in that order. The ratio $TR:PS$ is equal to

Question illustration
  1. $1:\sqrt3$
  2. $1:2\sqrt3$
  3. $1:2$
  4. $1:\sqrt2$
  5. $1:\frac{3\sqrt3}{2}$

Question 15

The solution to the equation $2017x^2-2017^{2017}=0$ is
  1. $x=\pm2017$
  2. $x=\pm2017^{2017}$
  3. $x=\pm2017^{1008}$
  4. $x=\pm2017^{1009}$
  5. $x=\pm\sqrt{2017^{2017}}$

Question 16

A hemispherical bowl is filled to half its depth. The maximum angle through which the bowl may be tilted from horizontal without spilling is

Question illustration
  1. $22\frac12^\circ$
  2. $30^\circ$
  3. $45^\circ$
  4. $60^\circ$
  5. $67\frac12^\circ$

Question 17

I spent exactly 4000 dollars on books which cost 25 dollars and 26 dollars each. I bought at least one book at each price. What is the maximum number of books I could have bought?
  1. 153
  2. 157
  3. 155
  4. 159
  5. 161

Question 18

All of the digits from 0 to 9 are used to form two 5-digit numbers. What is the smallest possible difference between these two numbers?
  1. 1
  2. 9
  3. 99
  4. 247
  5. 315

Question 19

Two circles of diameter 12 cm overlap, forming the shape shown with perimeter $14\pi$ cm. What is the area of the shape, in $\mathrm{cm}^2$?

Question illustration
  1. $42\pi+18$
  2. $42\pi+9\sqrt3$
  3. $72\pi-18$
  4. $72\pi-9\sqrt3$
  5. $24\pi\sqrt3$

Question 20

Five spherical balls of diameter 10 cm fit inside a closed cylindrical tin with internal diameter 16 cm. What is the smallest height the tin can be?
  1. 39 cm
  2. 42 cm
  3. 45 cm
  4. 48 cm
  5. 50 cm