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Australian AMC 2014 - Senior

Question 1

Which expression has the same meaning as $9x^{-3}$?
  1. $-9/x^3$
  2. $3/x$
  3. $1/(9x^3)$
  4. $3/x^3$
  5. $9/x^3$

Question 2

What is the value of $1/0.04$?
  1. 15
  2. 20
  3. 25
  4. 40
  5. 60

Question 3

If $p=9$ and $q=-3$, what is $p^2-q^2$?
  1. 64
  2. 72
  3. 84
  4. 90
  5. 96

Question 4

A circle has circumference $\pi$ units. What is its area, in square units?
  1. $\pi/4$
  2. $\pi/2$
  3. $\pi$
  4. $2\pi$
  5. $4\pi$

Question 5

If $K=L+6/R$, $L=4$ and $K=7$, what is $R$?
  1. -18
  2. 1
  3. 12
  4. 8
  5. 2

Question 6

If $x,x^2,x^3$ lie on a number line in the order shown, which could be $x$?

Question illustration
  1. -2
  2. $-1/2$
  3. $3/4$
  4. 1
  5. $3/2$

Question 7

A 2 metre broom leans against a wall, making $45^\circ$ with the ground at its bottom. It slowly slides down until this angle is $30^\circ$. How far, in metres, has the top slid down the wall?

Question illustration
  1. $\sqrt2-1$
  2. $2-\sqrt3$
  3. $\sqrt3-1$
  4. $\sqrt3-\sqrt2$
  5. $2-\sqrt2$

Question 8

The base of a triangle is increased by 25% and its height by 50%. By what percentage has its area increased?
  1. 25%
  2. 50%
  3. $66.\dot6\%$
  4. 75%
  5. 87.5%

Question 9

The five intervals are marked on the number line. If $x$ falls in interval A and $y$ in interval D, in which interval must $\tfrac12(x+y+1)$ fall?

Question illustration
  1. A
  2. B
  3. C
  4. D
  5. E

Question 10

If $p/(p-2q)=3$, what is $p/q$?
  1. -3
  2. 3
  3. $1/3$
  4. $2/3$
  5. 2

Question 11

A car park contains 3 Fords,4 Holdens and 2 Hondas. An inspector chooses 2 cars at random. What is the probability both are Holdens?
  1. $1/4$
  2. $4/27$
  3. $1/6$
  4. $4/9$
  5. $16/81$

Question 12

In the figure, $P,Q$ are the centres of two circles. Each circle has area $10\text{ m}^2$. What is the area of the rectangle in square metres?

Question illustration
  1. 20
  2. $20-10/\pi$
  3. $40/\pi$
  4. $50/\pi$
  5. $60/\pi$

Question 13

Between which two numbers does $\sqrt{1+2+3+\cdots+99+100}$ lie?
  1. 14 and 15
  2. 25 and 26
  3. 30 and 31
  4. 71 and 72
  5. 100 and 101

Question 14

If $(x-a)/(x-b)=(x-b)/(x-a)$ and $a\ne b$, what is $x$?
  1. $(a-b)/2$
  2. $(a^2+b^2)/(a+b)$
  3. $(a^2+b^2)/(2(a+b))$
  4. $a+b$
  5. $(a+b)/2$

Question 15

In the diagram, $PS=5$, $PQ=3$, triangle $PQS$ is right-angled at $Q$, $\angle QSR=30^\circ$, and $QR=RS$. What is $RS$?

Question illustration
  1. $\sqrt3/2$
  2. $\sqrt3$
  3. 2
  4. $4\sqrt3/3$
  5. 4

Question 16

Billy, a seasonal worker in Cowra, collected an even number of buckets of cherries on his first day. Each following day he picked 2 more buckets than the previous day. In the first 50 days he collected 3250 buckets. How many did he collect on day 50?
  1. 66
  2. 110
  3. 114
  4. 116
  5. 120

Question 17

A farmer walks 1 km east, then 1 km north-east, then another 1 km east. What is the distance in kilometres between his initial and final positions?
  1. $\sqrt{5+2\sqrt2}$
  2. $\sqrt{10}$
  3. $\sqrt5$
  4. $2+\sqrt2$
  5. $\sqrt{11/2+2\sqrt{10}}$

Question 18

Two machines move at constant speeds around a circle of circumference 600 cm, starting together at the same point. In the same direction they next meet after 20 seconds; in opposite directions they next meet after 5 seconds. What is the faster speed in centimetres per second?
  1. 60
  2. 65
  3. 70
  4. 75
  5. 85

Question 19

The equation $x^2-kx+374=0$ has two integer solutions. How many distinct values of $k$ are possible?
  1. 2
  2. 4
  3. 6
  4. 8
  5. 10

Question 20

Given $f_1(x)=x/(x+1)$ and $f_{n+1}(x)=f_1(f_n(x))$, what is $f_{2014}(x)$?
  1. $x/(2014x+1)$
  2. $2014x/(2014x+1)$
  3. $x/(x+2014)$
  4. $2014x/(x+1)$
  5. $x/(2014(x+1))$