AMC Solutions Book 2016

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It's not that I'm so

smart,

it's just that I stay

with

problems longer.

Albert EinsteinAlbert Einstein

How many integers

in e set 100, 101,

102,...,999 do n

conin e digs

1 or 2 or 3 or 4?

I have a $10 note and an ice-cream costs $2.20. What

is the greatest number of ice-creams I c

an buy?

(A)

3

3 (B)

(B)

4

4 (C)

(C)

5

5 (D)

(D)

6

6 (E)

(E)

7

a

2

_{+ b}

2

_{= c}

2

2 Pythagoras’ theorem

Pythagoras’ theorem

Algebra is the

language thr

ough

which we de

scribe

patterns.

A

A US USTRTRALALIAIANN MMA T H E M A T I C SA T H E M A T I C S TT RU RUSTST

ww www.w.amamtt.e.eddu.u.auau

maths

can

take

you

any

where

problem

solving

time distance speed

T

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S

T

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O

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L

F

monomial

monomial binomialbinomial trinomial trinomial

I have $10 in 10-cent coins, $10 in

20-cent coins and $10 in 50-cent

coins. How many coins do I

coins. How many coins do I have?

have?

If p = 11 and q = –4,

then p

22

– q

22

equals ...

2016 Solutions

A 40 x 40 whe square is divided in 1 x 1 squares

by lines parael  s sides. Some of ese 1 x 1

squares are coloured red so at each of e 1 x 1

squares, regardle of wheer  is coloured red

or n, shares a side wh at mo one red square

(n counng self). What is e large poible

number of red squares?

p

a

r

a

b

o

_l

a

pigeonhole principle

if n pigeons are put in m holes,

wh n > m, en

at lea one hole

mu conin more an one pigeon.

3 3 . . 1 1 4 4 1 1 5 5 9 9 2 2 6 6 5 5 3 3 5 5 8 8 9 9 7 7 9 933₂₂₃₃ 8 8 4 4 6 6 2 2 6 6 4 4 3 3 3 3 8 8 3 3 2 2 7 7 9 9

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2016 AMC 2016 AMC iiii

2016 Solutions

About the Australian Mathematics Competition

The Australian Mathematics C

The Australian Mathematics Competition (AMC) was introduced in Austrompetition (AMC) was introduced in Australia in 1978 asalia in 1978 as the ﬁrst Australia-wide mathematics competition for students. Since then it

the ﬁrst Australia-wide mathematics competition for students. Since then it has servedhas served almost all Australian secondary schools and

almost all Australian secondary schools and many primary schools, providing feedbackmany primary schools, providing feedback and enrichment to schools and

and enrichment to schools and students. A truly international event, there are entries fromstudents. A truly international event, there are entries from more than 30 countries across South-East

more than 30 countries across South-East Asia, the Paciﬁc, Europe, Africa and the MiddleAsia, the Paciﬁc, Europe, Africa and the Middle East. As of 2016, the AMC has attracted more than 14.75 million entries.

East. As of 2016, the AMC has attracted more than 14.75 million entries. The AMC is for students of all standar

The AMC is for students of all standards. Students are asked to solvds. Students are asked to solve 30 problems in 60e 30 problems in 60 minutes (Years 3–6) or 75 minutes (

minutes (Years 3–6) or 75 minutes ( YYears 7–12). The earliest problems are very easy. Allears 7–12). The earliest problems are very easy. All students should be able

students should be able to attempt them. The problems get progressively more diﬃcultto attempt them. The problems get progressively more diﬃcult until the end, when they are

until the end, when they are challenging to the most gifted student. challenging to the most gifted student. Students of allStudents of all standards will make progress and ﬁnd

standards will make progress and ﬁnd a point of challenge.a point of challenge. The AMC is a fun competition with many of the

The AMC is a fun competition with many of the problems set in situations familiar toproblems set in situations familiar to students and showing the

students and showing the relevance of mathematics in their everyday lives. The problemsrelevance of mathematics in their everyday lives. The problems are also designed to stimulate discussion and can be used by teachers and students as are also designed to stimulate discussion and can be used by teachers and students as springboards for investigation.

springboards for investigation. There are ﬁv

There are ﬁve papers: Middle Primary (Years 3–4), Uppe papers: Middle Primary (Years 3–4), Upper Primary (Years 5–6), Jer Primary (Years 5–6), Junior (Yearsunior (Years 7–8), Intermediate (Years 9–10) and Senior (

7–8), Intermediate (Years 9–10) and Senior (YYears 11–12). Questions 1–10 are worthears 11–12). Questions 1–10 are worth 3 marks each, questions 11–20 are worth 4 marks, questions 21–25 are worth 5 marks, 3 marks each, questions 11–20 are worth 4 marks, questions 21–25 are worth 5 marks, while questions 26–30 are valued at

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2016 AMC – Middle Primary Questions 2016 AMC – Middle Primary Questions

Questions – Middle Primary Division

1.

1. What is the vWhat is the value of 20 alue of 20 + + 16?16?

((AA) ) 224 4 ((BB) ) 226 6 ((CC) ) 336 6 ((DD) ) 9 9 ((EE) ) 221166

2.

2. WhicWhich of these nh of these numbers is umbers is the smallthe smallest?est?

((AA) ) 66555 5 ((BB) ) 55666 6 ((CC) ) 55665 5 ((DD) ) 55555 5 ((EE) ) 555566

3.

3. In the nuIn the number 83mber 83 014, th014, the digie digit 3 represt 3 represenentsts

((AA) ) tthhrreee e ((BB) ) tthhiirrtty y ((CC) ) tthhrreee e hhuunnddrreedd ((DD) ) tthhrreee e tthhoouussaannd d ((EE) ) tthhiirrtty y tthhoouussaanndd

4.

4. My sistMy sister is 6 years oler is 6 years old and I am d and I am twtwice her ageice her age. . AddAdding our ageing our ages givs giveses

((AA) ) 114 4 ((BB) ) 115 5 ((CC) ) 118 8 ((DD) ) 220 0 ((EE) ) 2211

5.

5. FFour of these shapes havour of these shapes have one or more lines of symmetrye one or more lines of symmetry. . WhicWhich one does not?h one does not?

((AA) ) ((BB) ) ((CC) ) ((DD) ) ((EE))

6.

6. TwTwo pizzas are slio pizzas are sliced inced into quarteto quarters. rs. HoHow many sliw many slices wilces will therel there be?

be?

((AA) ) 2 2 ((BB) ) 110 0 ((CC) ) 66 ((DD) ) 8 8 ((EE) ) 1166

7.

7. Will has a 45-minWill has a 45-minute music lesute music lesson every Tson every Tuesdauesday afternoon after school. y afternoon after school. If it beginsIf it begins at 4:30

at 4:30 pm, at what time does it ﬁnispm, at what time does it ﬁnish?h?

((AA) ) 44::445 p5 pm m ((BB) ) 44::555 p5 pm m ((CC) ) 44::775 p5 pm m ((DD) ) 55::000 p0 pm m ((EE) ) 55::115 p5 pmm

1

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2016

2016 AMC – Middle AMC – Middle Primary QuestionsPrimary Questions

8.

8. _{In our garage th}_{In our garage there are 4 bicycl}_{ere are 4 bicycles, 2 tricy}_{es, 2 tricycles and one}_{cles and one}

quad bike. How many wheels are there altogether? quad bike. How many wheels are there altogether? ((AA) ) 3 3 ((BB) ) 6 6 ((CC) ) 77

((DD) ) 114 4 ((EE) ) 1188

9.

9. _T_{Ten chai}_{en chairs are equal}_{rs are equally spac}_{ly spaced around a round tabl}_{ed around a round table.}_{e. They are nu}_{They are numbe}_{mbered 1}_{red 1 to 10 in}_{to 10 in}

order. Which chair is opposite chair 9? order. Which chair is opposite chair 9?

((AA) ) 1 1 ((BB) ) 2 2 ((CC) ) 3 3 ((DD) ) 4 4 ((EE) ) 55

10.

10. _Lee’_{Lee’s favo}_{s favouri}_{urite chocol}_{te chocolates are 80c}_{ates are 80c eac}_each._{h. He has}_{He has ﬁv}_ﬁve_{e dol}_{dollars to}_{lars to}

spend. How many of these chocolates can he buy? spend. How many of these chocolates can he buy?

((AA) ) 4 4 ((BB) ) 5 5 ((CC) ) 6 6 ((DD) ) 7 7 ((EE) ) 88

11.

11._{The four digits 2, 3, 8 and 9 are placed in the boxes}_{The four digits 2, 3, 8 and 9 are placed in the boxes}

so that when both two-digit numbers are added, the so that when both two-digit numbers are added, the sum is as large as possible. What is this sum?

sum is as large as possible. What is this sum?

((AA) ) 11775 5 ((BB) ) 667 7 ((CC) ) 115566 ((DD) ) 11779 9 ((EE) ) 112211

+ +

12.

12._{A circular piece of paper is folded in half twice and then a cut is made as shown.}_{A circular piece of paper is folded in half twice and then a cut is made as shown.}

When the piece of paper is unfolded, what shape is the hole in the centre? When the piece of paper is unfolded, what shape is the hole in the centre?

((AA) ) ((BB) ) ((CC) ) ((DD) ) ((EE))

2

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13.

13._{Phoebe put}_{Phoebe put he}_her_{r han}_hand_{d in her}_{in her pock}_pocket_{et an}_and_{d pul}_pulle_led_{d out 60}_{out 60 cen}_cents_{ts. . Ho}_How_{w ma}_{many diﬀ}_{ny diﬀere}_erent_nt

ways could this amount be made using 10c, 20c and 50c coins? ways could this amount be made using 10c, 20c and 50c coins?

((AA) ) 2 2 ((BB) ) 3 3 ((CC) ) 4 4 ((DD) ) 5 5 ((EE) ) 66

14.

14._{There are 5 red, 5 green and 5 yellow jelly beans in a jar.}_{There are 5 red, 5 green and 5 yellow jelly beans in a jar.}

How many would you need to take out of the jar without How many would you need to take out of the jar without looking to make sure that you have removed at least two of looking to make sure that you have removed at least two of the same colour?

the same colour?

((AA) ) 3 3 ((BB) ) 4 4 ((CC) ) 55 ((DD) ) 6 6 ((EE) ) 77

15.

15._{A sailor coiled a rope on his ship’s deck, and}_{A sailor coiled a rope on his ship’s deck, and}

some pai

some paint want was spills spilled across haled across half of f of it. it. WhatWhat did the rope look like when it was uncoiled? did the rope look like when it was uncoiled?

(A) (A) (B) (B) (C) (C) (D) (D) (E) (E) 16.

16._{The students in Mr Day’s class were asked}_{The students in Mr Day’s class were asked}

the colou

the colour of r of theitheir sun r sun hat. hat. The resulThe results arets are shown in the graph.

shown in the graph.

Mr Day chooses two colours which include Mr Day chooses two colours which include the hat colours of exactly half of the class. the hat colours of exactly half of the class. Which two colours does he choose?

Which two colours does he choose? (A) orange and black

(A) orange and black (B)

(B) greegreen and yelln and yellowow (C) black and yellow (C) black and yellow (D) red and orange (D) red and orange (E)

(E) red and yred and yellellowow

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7

rreed d oorraanngge e bbllaacck k ggrreeeen n yyeellllooww Sun hat colours

Sun hat colours

17.

17. _{The sum of}_{The sum of the sev}_{the seven digits in Mario}_{en digits in Mario’s teleph}_{’s telephone num}_{one number is}_{ber is 34.}_{34. The first five digi}_{The first five digits}_ts

are 73903. How many possibilities are there for the last two digits? are 73903. How many possibilities are there for the last two digits?

((AA) ) 6 6 ((BB) ) 7 7 ((CC) ) 8 8 ((DD) ) 9 9 ((EE) ) 1100

3

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2016

18.

18._{If the area of the tangram shown is 64 square cen-}_{If the area of the tangram shown is 64 square}

cen-timetres, what is the area in square centimetres of timetres, what is the area in square centimetres of the small square?

the small square?

((AA) ) 332 2 ((BB) ) 224 4 ((CC) ) 1166 ((DD) ) 8 8 ((EE) ) 44

19.

19. _{By making just one fold on a}_{By making just one fold on a rectangul}_{rectangular piece of paper,}_{ar piece of paper, whic}_{which of the}_{h of the follo}_{following shapes}_{wing shapes}

is NOT possible? is NOT possible?

((AA)) ((BB)) ((CC))

((DD)) ((EE))

20.

20._{In this diagram there are four lines with three circles each.}_{In this diagram there are four lines with three circles each.}

Place the numbers from 1 to 7 into the circles, so that each Place the numbers from 1 to 7 into the circles, so that each lin

line adds up e adds up to 12. to 12. WhiWhich numch number must go ber must go ininto the to the circirclecle at the centre of the diagram?

at the centre of the diagram?

((AA) ) 7 7 ((BB) ) 6 6 ((CC) ) 5 5 ((DD) ) 4 4 ((EE) ) 22

21.

21. _F_{Four hock}_{our hockey teams pla}_{ey teams play}_{y eac}_each_{h of the}_{of the othe}_other_{r thre}_three_{e team}_teams_{s once}_{once. . A win}_{A win sco}_{scores 3}_{res 3 poin}_points,_ts,

a draw scores 1 point and a loss scores 0 points. Some ﬁgures in the following table a draw scores 1 point and a loss scores 0 points. Some ﬁgures in the following table are missing. How many points did the Hawks get?

are missing. How many points did the Hawks get? Pl

Plaayyeed d WWiin n DDraraw w LoLosss s PPoioinntsts E Eaaggllees s 3 3 3 3 99 H Haawwkks s 33 F Faallccoonns s 3 3 0 0 11 C Coonnddoorrs s 3 3 0 0 2 2 11 ((AA) ) 1 1 ((BB) ) 4 4 ((CC) ) 6 6 ((DD) ) 7 7 ((EE) ) 1100 4 4

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22.

22._{In this grid you can only move downward, going from point}_{In this grid you can only move downward, going from point}

to point along the lines shown. to point along the lines shown. One route from

One route from P P toto QQ is drawn in. is drawn in.

How many diﬀerent routes are there from

How many diﬀerent routes are there from P P toto QQ??

((AA) ) 2 2 ((BB) ) 4 4 ((CC) ) 66 ((DD) ) 8 8 ((EE) ) 1122 P P Q Q 23.

23._{I have ﬁve coloured discs in a pile as shown.}_{I have ﬁve coloured discs in a pile as shown.}

I take the top two discs and put them on the bottom I take the top two discs and put them on the bottom (with the red disc still on top of the blue disc).

(with the red disc still on top of the blue disc).

Then I again take the top two discs and put them on the Then I again take the top two discs and put them on the bottom.

bottom.

If I do this until I have made a total of 21 moves, which If I do this until I have made a total of 21 moves, which disc will be on the bottom?

disc will be on the bottom?

orange orange yellow yellow green green blue blue red red

((AA) ) rreed d ((BB) ) bblluue e ((CC) ) ggrreeeen n ((DD) ) yyeelllloow w ((EE) ) oorraannggee

24.

24. _{A zoo keeper weighed some of the animals at Melbourne}_{A zoo keeper weighed some of the animals at Melbourne}

Zoo.

Zoo. He founHe found that the lion wd that the lion weigeighs 90hs 90 kg more thakg more than then the leo

leopardpard, , and the tiger weigand the tiger weighs 50hs 50 kg less than the kg less than the liolion.n. Alt

Altogetogether the her the threthree e animanimals weials weigh 310gh 310 kg. kg. HoHow w mumuchch does the lion weigh?

does the lion weigh?

((AA) ) 11880 k0 kg g ((BB) ) 11550 k0 kg g ((CC) ) 11440 k0 kg g ((DD) ) 11330 k0 kg g ((EE) ) 11000 k0 kgg

25.

25._{Jane and Tom each have $3.85 in coins, one of each Australian coin. They each give}_{Jane and Tom each have $3.85 in coins, one of each Australian coin. They each give}

some coins to Angus so that Tom has exactly twice as much money as Jane. some coins to Angus so that Tom has exactly twice as much money as Jane. What is the smallest number of coins given to Angus?

What is the smallest number of coins given to Angus?

55 cents cents 10 10 cents cents 20 20 cents cents

50

cents cents 11 dollar dollar 22 dollars dollars ((AA) ) 2 2 ((BB) ) 3 3 ((CC) ) 4 4 ((DD) ) 6 6 ((EE) ) 88 5 5 5 5

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2016

26.

26._{With some 3-dig}_{With some 3-digit num}_{it numbers}_bers,_{, the thi}_{the third digit is}_{rd digit is the sum of}_{the sum of the ﬁrst tw}_{the ﬁrst two}_{o dig}_digits_its._{. F}_For_or

example, with the number 213 we can add 1 and 2 to get 3, so the third digit is the example, with the number 213 we can add 1 and 2 to get 3, so the third digit is the sum of the ﬁrst two digits.

sum of the ﬁrst two digits.

How many 3-digit numbers are there where the third digit is the sum of the ﬁrst two How many 3-digit numbers are there where the third digit is the sum of the ﬁrst two digits?

digits?

27.

27._{In a family with two sons and two daughters, the sum of the children’s ages is 55.}_{In a family with two sons and two daughters, the sum of the children’s ages is 55.}

The two sons were born three years apart, and the two daughters were born two years The two sons were born three years apart, and the two daughters were born two years apart. The younger son is twice the age of the older daughter.

apart. The younger son is twice the age of the older daughter. How old is the youngest child?

How old is the youngest child?

28.

28. _F_{From this set}_{rom this set of six stamps,}_{of six stamps, how many wa}_{how many ways could you}_{ys could you}

ch

choose oose thrthree ee stastamps mps thathat t are are conconnecnected ted aloalong ng thetheirir edges? edges? C D E F C D E F A A BB 29.

29._A_{A cla}_class_{ss has 2016 matc}_{has 2016 matchst}_hstic_icks._{ks. Usi}_{Using blobs of}_{ng blobs of mode}_modelli_lling_{ng cla}_clay_{y to}_{to joi}_join_{n the matc}_{the matche}_hess

together, they make a long row of cubes. This is how their row starts. together, they make a long row of cubes. This is how their row starts.

They keep adding cubes to the end of the row until they don’t have enough matches They keep adding cubes to the end of the row until they don’t have enough matches left for another cube. How many cubes will they make?

left for another cube. How many cubes will they make?

30.

30._{Mary has four children of diﬀerent ages, all under 10, and the product of their ages}_{Mary has four children of diﬀerent ages, all under 10, and the product of their ages}

is 2016. What is the sum of their ages? is 2016. What is the sum of their ages?

6

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2016 AMC – Upper Primary Questions

2016 AMC – Upper Primary Questions 7777

Questions – Upper Primary Division

1.

1. WhicWhich of these nh of these numbers iumbers is the smalls the smallest?est?

((AA) ) 66555 5 ((BB) ) 55666 6 ((CC) ) 55665 5 ((DD) ) 55555 5 ((EE) ) 555566

2.

2. TwTwo pizzas are slio pizzas are sliced intced into quartero quarters. s. HoHow many sliw many slices wilces will therel there be?

be?

((AA) ) 2 2 ((BB) ) 110 0 ((CC) ) 66 ((DD) ) 8 8 ((EE) ) 1166

3.

3. Join Join the the dotsdots P P ,, QQ,, RR to form the triangle to form the triangle PP QRQR..

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • P P QQ R R

How many dots lie

How many dots lie inside inside _{the triangle}_{the triangle} _P_{P QR}_QR_??

((AA) ) 113 3 ((BB) ) 114 4 ((CC) ) 115 5 ((DD) ) 117 7 ((EE) ) 1188

4.

4. 00..3 + 03 + 0..4 4 isis

((AA) ) 00..007 7 ((BB) ) 00..7 7 ((CC) ) 00..112 2 ((DD) ) 00..1 1 ((EE) ) 77

5.

5. Lee’Lee’s favs favouriourite chocolte chocolates are 80c eacates are 80c each. h. He has ﬁve dolHe has ﬁve dollars tolars to spend. How many of these chocolates can he buy?

spend. How many of these chocolates can he buy?

((AA) ) 4 4 ((BB) ) 5 5 ((CC) ) 6 6 ((DD) ) 7 7 ((EE) ) 88

6.

6. TTen chaien chairs are equalrs are equally spacly spaced around a round tabled around a round table. e. They are nuThey are numbembered 1 red 1 to 10 into 10 in order. Which chair is opposite chair 9?

order. Which chair is opposite chair 9?

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2016 AMC – Upper Primary Questions 2016 AMC – Upper Primary Questions 8

8

7.

7. _In_{In a}_{a pi}_piece_{ece of}_{of mu}_musi_sic,_{c, a}_{a not}_note_{e li}_like_{ke is}_{is wo}_worth_{rth on}_one_{e bea}_beat,_t, _is_{is wo}_worth_{rth ha}_half_{lf a}_{a bea}_beat,_{t, is}_{is wo}_worth_rth

2 beats and

2 beats and is wis worth 4 beatorth 4 beats. s. HoHow manw many beats ary beats are in the fole in the follolowinwing piecg piece of muse of music?ic?

((AA) ) 4 4 ((BB) ) 5 5 ((CC) ) 6 6 ((DD) ) 7 7 ((EE) ) 88

8.

8. _Ph_{Phoebe put her hand in her}_{oebe put her hand in her poc}_pocke_ket_{t and pul}_{and pulle}_led_{d ou}_out_{t 60 cen}_{60 cents.}_{ts. Ho}_How_{w ma}_{many di}_{ny diﬀe}_ﬀeren_rentt

ways could this amount be made using 10c, 20c and 50c coins? ways could this amount be made using 10c, 20c and 50c coins?

((AA) ) 2 2 ((BB) ) 3 3 ((CC) ) 4 4 ((DD) ) 5 5 ((EE) ) 66

9.

9. _Whic_{Which of these contain}_{h of these containers is current}_{ers is currently holding the most w}_{ly holding the most water?}_ater?

(A) (A) 30030000 mLmL _(B)_(B) 100 10000 mLmL (C) (C) 1000mL 1000mL (D) (D) 75 7500 mLmL (E) (E) 20020000 mLmL 10.

10._{Which of these shapes has the most axes of symmetry (mirror lines)?}_{Which of these shapes has the most axes of symmetry (mirror lines)?}

((AA)) ((BB)) ((CC))

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2016 AMC – Upper Primary Questions

2016 AMC – Upper Primary Questions 9999

11.

11._{A sailor coiled a rope on his ship’s deck, and}_{A sailor coiled a rope on his ship’s deck, and}

som

some paint wae paint was spills spilled across haled across half of f of it. it. WhatWhat did the rope look like when it was uncoiled? did the rope look like when it was uncoiled?

(A) (A) (B) (B) (C) (C) (D) (D) (E) (E) 12.

12._{If the area of the tangram shown is 64 square cen-}_{If the area of the tangram shown is 64 square}

cen-timetres, what is the area in square centimetres of timetres, what is the area in square centimetres of the small square?

the small square?

((AA) ) 332 2 ((BB) ) 224 4 ((CC) ) 1166 ((DD) ) 8 8 ((EE) ) 44

13.

13._{For each batch of 25 biscuits, Jack uses 2}_{For each batch of 25 biscuits, Jack uses 2}11

2

2 pack packets of ets of chchocolaocolate te chchipsips. . HoHow w manmanyy

packets does he need if he wants to bake 200 biscuits? packets does he need if he wants to bake 200 biscuits?

((AA) ) 220 0 ((BB) ) 8 8 ((CC) ) 880 0 ((DD) ) 110 0 ((EE) ) 5500

14.

14._{Which one of the following is correct?}_{Which one of the following is correct?}

(A) Two even numbers add to an odd number. (A) Two even numbers add to an odd number. (B)

(B) An odd An odd nunumbember minus an odd r minus an odd nunumbember is r is alwalwayays odd.s odd. (C)

(C) AddAdding 2 ing 2 odd numbeodd numbers and rs and an even numan even number is ber is alwalwayays os odd.dd. (D) Adding 3 odd numbers is always odd.

(D) Adding 3 odd numbers is always odd. (E)

(E) An odd number multiAn odd number multiplied by an odd number alwaplied by an odd number always equals an even numys equals an even number.ber.

15.

15._{The perimeter of the outer square is 36cm, and the}_{The perimeter of the outer square is 36cm, and the}

perim

perimeter of the eter of the inninner squarer square is e is 2020 cm.cm.

If the four rectangles are all identical, what is the If the four rectangles are all identical, what is the perime-ter of the shaded rectangle in centimetres?

ter of the shaded rectangle in centimetres?

((AA) ) 112 2 ((BB) ) 114 4 ((CC) ) 2244 ((DD) ) 220 0 ((EE) ) 1188

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10

16.

16._{George has a new lock that opens if the four numbers 1, 2, 3 and}_{George has a new lock that opens if the four numbers 1, 2, 3 and}

4 are pressed once each in the correct order. 4 are pressed once each in the correct order.

If the ﬁrst number must be larger than the second number, how If the ﬁrst number must be larger than the second number, how many combinations are possible?

many combinations are possible?

((AA) ) 110 0 ((BB) ) 112 2 ((CC) ) 1155 ((DD) ) 118 8 ((EE) ) 2200 3 3 1 1 4 4 2 2 17.

17._{A straight cut is made through the hexagon shown to}_{A straight cut is made through the hexagon shown to}

crea

create te twtwo o new shapes. new shapes. WhiWhich of ch of the follothe followinwing g coulcouldd

not

not_{be made?}_{be made?}

(A) one triangle and one hexagon (A) one triangle and one hexagon (B)

(B) twtwo pentagono pentagonss (C)

(C) twtwo quadrilato quadrilateralserals

(D) one quadrilateral and one pentagon (D) one quadrilateral and one pentagon (E)

(E) one trianone triangle and one gle and one quadquadrilrilaterateralal

18.

18._{The numbers 3, 9, 15, 18, 24 and 29 are divided into two groups of 3 numbers and}_{The numbers 3, 9, 15, 18, 24 and 29 are divided into two groups of 3 numbers and}

eac

each h grougroup p is added. is added. The differThe difference betwence between the een the twtwo o sums (totasums (totals) of ls) of 3 3 nunumbembers isrs is as small as possible. What is the smallest difference?

as small as possible. What is the smallest diﬀerence?

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 5 5 ((EE) ) 88

19.

19._{Benny built a magic square using the numbers from 1}_{Benny built a magic square using the numbers from 1}

to 16, where the numbers in each row, each column and to 16, where the numbers in each row, each column and each diagonal add up to the same total.

each diagonal add up to the same total. What number does he place at the X? What number does he place at the X?

((AA) ) 116 6 ((BB) ) 115 5 ((CC) ) 117 7 ((DD) ) 111 1 ((EE) ) 1144 ₄₄ ₁₁ 7 7 1212 5 5 1010 X X ₁₃₁₃ 20.

20._{Andy has a number of red, green and blue counters.}_{Andy has a number of red, green and blue counters.}

He places eight counters equally spaced around a circle He places eight counters equally spaced around a circle ac-cording to the following rules:

cording to the following rules:

•

• No two red counters will be next to each other. No two red counters will be next to each other. •

• No two green counters will be diagonally opposite each No two green counters will be diagonally opposite each

other. other.

•

• As few blue counters as possible will be used. As few blue counters as possible will be used.

How many blue counters will Andy need to use? How many blue counters will Andy need to use?

(14)

2016 AMC – Junior Questions

2016 AMC – Junior Questions 11111111

21.

21._{I have ﬁve coloured discs in a pile as shown.}_{I have ﬁve coloured discs in a pile as shown.}

I take the top two discs and put them on the bottom I take the top two discs and put them on the bottom (with the red disc still on top of the blue disc).

(with the red disc still on top of the blue disc).

Then I again take the top two discs and put them on the Then I again take the top two discs and put them on the bottom.

bottom.

If I do this until I have made a total of 21 moves, which If I do this until I have made a total of 21 moves, which disc will be on the bottom?

disc will be on the bottom?

orange orange yellow yellow green green blue blue red red

((AA) ) rreed d ((BB) ) bblluue e ((CC) ) ggrreeeen n ((DD) ) yyeelllloow w ((EE) ) oorraannggee

22.

22. _{A zoo keeper weighed some of the animals at Melbourne}_{A zoo keeper weighed some of the animals at Melbourne}

Zoo.

Zoo. He founHe found that the lion wd that the lion weigeighs 90hs 90 kg more thakg more than then the leo

leopardpard, , and the tiger weigand the tiger weighs 50hs 50 kg less than the kg less than the liolion.n. Alt

Altogetogether the her the threthree e animanimals weials weigh 310gh 310 kg. kg. HoHow w mumuchch does the lion weigh?

does the lion weigh?

((AA) ) 11880 k0 kg g ((BB) ) 11550 k0 kg g ((CC) ) 11440 k0 kg g ((DD) ) 11330 k0 kg g ((EE) ) 11000 k0 kgg

23.

23._{Adrienne, Betty and Cathy were the only three competitors participating in a series}_{Adrienne, Betty and Cathy were the only three competitors participating in a series}

of athletic events. In each event, the winner gets 3 points, second gets 2 points and of athletic events. In each event, the winner gets 3 points, second gets 2 points and third gets 1 point. After the events, Adrienne has 8 points, Betty has 11 points and third gets 1 point. After the events, Adrienne has 8 points, Betty has 11 points and Cathy has 5 points. In how many events did Adrienne come second?

Cathy has 5 points. In how many events did Adrienne come second?

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 3 3 ((EE) ) 44

24.

24._{Jane and Tom are comparing their pocket money. Jane has as many 5c coins as Tom}_{Jane and Tom are comparing their pocket money. Jane has as many 5c coins as Tom}

has 10c coins and as

has 10c coins and as manmany 10c y 10c coicoins as ns as TTom has om has 20c coins20c coins. . HoHowewevever, Jane has r, Jane has asas many 50c coins as Tom has 5c coins.

many 50c coins as Tom has 5c coins.

They have no other coins and they ﬁnd that they each have the same amount of They have no other coins and they ﬁnd that they each have the same amount of money.

money.

What is the smallest number of coins they each can have? What is the smallest number of coins they each can have?

((AA) ) 3 3 ((BB) ) 4 4 ((CC) ) 5 5 ((DD) ) 6 6 ((EE) ) 77

25.

25._{A tuckshop has two jars of cordial mixture.}_{A tuckshop has two jars of cordial mixture.}

Jar A is 30% cordial, while Jar B is 60% cordial. Jar A is 30% cordial, while Jar B is 60% cordial.

Some of Jar A is mixed with some of Jar B to make 18 litres of Some of Jar A is mixed with some of Jar B to make 18 litres of 50% cordial.

50% cordial.

How many litres from Jar A are used? How many litres from Jar A are used?

((AA) ) 9 9 ((BB) ) 112 2 ((CC) ) 44 ((DD) ) 3 3 ((EE) ) 66

(15)

12

26.

26._Qian_Qiang,_{g, Ror}_Rory_{y an}_and_{d Sop}_{Sophia are eac}_{hia are each}_{h w}_wear_{earing a}_{ing a hat wit}_{hat with}_{h a}_{a nu}_numbe_mber_{r on it.}_{on it. Eac}_Each_{h add}_addss

the tw

the two numbeo numbers on rs on the other twthe other two hats, giving totalo hats, giving totals of s of 11, 17 and 11, 17 and 22. 22. What is theWhat is the largest number on a hat?

largest number on a hat?

27.

27. _{The num}_{The number 840 is the 3-digit number with the most facto}_{ber 840 is the 3-digit number with the most factors.}_{rs. How man}_{How many factors does}_{y factors does}

it have? it have?

28.

28._A_{A cla}_class_{ss has 2016 matc}_{has 2016 matchst}_hstic_icks._{ks. Usi}_{Using blobs of}_{ng blobs of mode}_modelli_lling_{ng cla}_clay_{y to}_{to joi}_join_{n the matc}_{the matche}_hess

together, they make a long row of cubes. This is how their row starts. together, they make a long row of cubes. This is how their row starts.

They keep adding cubes to the end of the row until they don’t have enough matches They keep adding cubes to the end of the row until they don’t have enough matches left for another cube. How many cubes will they make?

left for another cube. How many cubes will they make?

29.

29._{You have an unlimited supply of five different coloured}_{You have an unlimited supply of five different coloured}

pop-sticks, and want to make

pop-sticks, and want to make as many diﬀerent colouredas many diﬀerent coloured equilateral triangles as possible, using three sticks. equilateral triangles as possible, using three sticks. One example is shown here.

One example is shown here.

Two triangles are not considered different if they are Two triangles are not considered different if they are rotations or reflections of each other.

rotations or reﬂections of each other.

How many diﬀerent triangles are possible?

How many diﬀerent triangles are possible? _blue_blue

r r e e d d r r e e d d 30.

30. _T_{Today my three cousins multiplie}_{oday my three cousins multiplied their ages together and it}_{d their ages together and it came to 2016.}_{came to 2016. This day}_{This day}

last year their ages multiplied to 1377. last year their ages multiplied to 1377.

When they multiplied their ages together 2 years ago today, what was their answer? When they multiplied their ages together 2 years ago today, what was their answer?

(16)

Questions – Junior Division

1.

1. The The vvalue alue of 20of 201616××2 2 isis

((AA) ) 4400226 6 ((BB) ) 4422112 2 ((CC) ) 4400222 2 ((DD) ) 44332 2 ((EE) ) 44003322

2.

2. In the diIn the diagram, thagram, the ve value of alue of xx isis

((AA) ) 330 0 ((BB) ) 220 0 ((CC) ) 9900 ((DD) ) 11440 0 ((EE) ) 110000 2020 ◦ ◦ 2020◦◦ x x◦◦ 3.

3. TToday is Thuoday is Thursdayrsday. . What day of the week wiWhat day of the week will it be 30 days from today?ll it be 30 days from today?

((AA) ) SSuunnddaay y ((BB) ) MMoonnddaay y ((CC) ) TTuueessddaay y ((DD) ) FFrriiddaay y ((EE) ) SSaattuurrddaayy

4.

4. TToday in Berracoday in Berracan, the miniman, the minimum temperature wum temperature wasas−−55◦◦C and the maximum was 8C and the maximum was 8◦◦CC

warmer than this. What was the maximum temperature? warmer than this. What was the maximum temperature? (A)

(A) −−33◦◦C C ((BB) ) 88◦◦C C ((CC)) −−1313◦◦C C ((DD) ) 1133◦◦C C ((EE) ) 33◦◦CC

5.

5. What What is is 25% 25% of of 11 22?? (A)

(A) 11₈₈ (B)(B) 11₄₄ (C)(C) 11₂₂ ((DD) ) 2 2 ((EE) ) 11

6.

6. A circuA circular pieclar piece of e of paper is foldpaper is folded in half twied in half twice and then a ce and then a cut is made as shocut is made as shown.wn.

When the piece of paper is unfolded, what shape is the hole in the centre? When the piece of paper is unfolded, what shape is the hole in the centre?

((AA) ) ((BB) ) ((CC) ) ((DD) ) ((EE))

7.

7. I used a $100 note to pay for a $29 book, a $16 calculI used a $100 note to pay for a $29 book, a $16 calculator and a pacator and a packeket of pens fort of pens for $8.95. What change did I get?

$8.95. What change did I get?

(17)

14

8.

8. _Whic_{Which of the followi}_{h of the following num}_{ng numbers is between 0.08 and 0.4?}_{bers is between 0.08 and 0.4?}

((AA) ) 00..00119 9 ((BB) ) 00..00009 9 ((CC) ) 00..11009 9 ((DD) ) 00..991 1 ((EE) ) 00..440099

9.

9. _{The cycli}_{The cycling road race throu}_{ng road race through the Adela}_{gh the Adelaide Hill}_{ide Hills started at 11:5}_{s started at 11:500 am and the winn}_{am and the winner}_er

took 74 minutes. The winner crossed the ﬁnishing line at took 74 minutes. The winner crossed the ﬁnishing line at

((AA) ) 11::2244 ppm m ((BB) ) 1122::5544 ppm m ((CC) ) 1122::0044 ppmm ((DD) ) 11::0044 ppm m ((EE) ) 1122::2244 ppmm

10.

10._{The fraction}_{The fraction} 720163 720163

2016 2016 isis

((AA) ) bebettwweeeen n 0 0 aannd d 1 1 ((BB) ) bebettwweeeen n 1 1 aannd d 110 0 ((CC) ) bbeettwweeeen n 110 0 aannd d 110000 ((DD) ) bbeettwweeeen n 11000 0 aannd d 1100000 0 ((EE) ) ggrreeaatteer r tthhaan n 11000000

11.

11._{The three squares shown have side lengths 3, 4}_{The three squares shown have side lengths 3, 4}

and 5.

and 5. What percentaWhat percentage of the area of the largestge of the area of the largest square is shaded?

square is shaded?

((AA) ) 2277% % ((BB) ) 2288% % ((CC) ) 2255%% ((DD) ) 2244% % ((EE) ) 2200%%

12.

12._{Jan has three times as many marbles as Liana. If Jan gives 3 of her marbles to Liana,}_{Jan has three times as many marbles as Liana. If Jan gives 3 of her marbles to Liana,}

they will have the same number. How many marbles do they have between them? they will have the same number. How many marbles do they have between them? ((AA) ) 118 8 ((BB) ) 6 6 ((CC) ) 8 8 ((DD) ) 112 2 ((EE) ) 1166

13.

13._{One of the pedestrian walkways in Hyde Park is}_{One of the pedestrian walkways in Hyde Park is}

exactly 3

exactly 311₂₂ sandstone pavers wide. The pavers are sandstone pavers wide. The pavers are arranged as shown.

arranged as shown.

The information sign says that 1750 pavers were The information sign says that 1750 pavers were used to make the walkway. How many pavers were used to make the walkway. How many pavers were cut in half in the construction of this walkway? cut in half in the construction of this walkway? ((AA) ) 22550 0 ((BB) ) 33550 0 ((CC) ) 117755

(18)

14.

14. _{On Monda}_{On Monday}_{y, , I}_{I pla}_plant_{nted 10}_{ed 10 appl}_apple_{e tree}_trees_{s in a}_{in a row}_{row. . On Tu}_{On Tuesd}_esday_{ay, , I}_{I pla}_plant_{nted orange trees}_{ed orange trees}

along the same row and noticed at the end of the day that no apple tree was next to along the same row and noticed at the end of the day that no apple tree was next to an apple tree. On Wednesday, I planted peach trees along the same row and noticed an apple tree. On Wednesday, I planted peach trees along the same row and noticed at the end of

at the end of ththe e daday y thathat t no applno apple e tretree e wawas next to s next to an oranan orange treege tree. . WhaWhat t is theis the smallest number of trees that I could have planted?

smallest number of trees that I could have planted?

((AA) ) 228 8 ((BB) ) 443 3 ((CC) ) 337 7 ((DD) ) 440 0 ((EE) ) 3366

15.

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 3 3 ((EE) ) 44

16.

16._{In the expression below, the letters}_{In the expression below, the letters}_A_{A,, B,C,D}_B,C,D _and_and _E_E_{represent the numbers 1}_{represent the numbers 1}_,,₂₂_,,₃₃_,,₄₄

and 5 in some order. and 5 in some order.

A

A××BB ++ C C ××DD ++E E

What is the largest possible value of the expression? What is the largest possible value of the expression?

((AA) ) 224 4 ((BB) ) 227 7 ((CC) ) 226 6 ((DD) ) 551 1 ((EE) ) 2255

17.

17._{Llewellyn uses four of these L-shaped tiles plus one other tile to completely cover a}_{Llewellyn uses four of these L-shaped tiles plus one other tile to completely cover a}

5 by 5 grid without any overlaps. 5 by 5 grid without any overlaps.

Which one of the following could be the other tile? Which one of the following could be the other tile?

(19)

16

18.

18._{Andy has a number of red, green and blue counters.}_{Andy has a number of red, green and blue counters.}

He places eight counters equally spaced around a circle He places eight counters equally spaced around a circle according to the following rules:

according to the following rules:

•

• No two red counters will be next to each other. No two red counters will be next to each other. •

• No two green counters will be diagonally opposite No two green counters will be diagonally opposite

each other. each other.

•

• As few blue counters as possible will be used. As few blue counters as possible will be used.

How many blue counters will Andy need to use? How many blue counters will Andy need to use?

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 3 3 ((EE) ) 44

19.

19._{In a packet of spaghetti, one-third of the strands of spaghetti are intact, but the}_{In a packet of spaghetti, one-third of the strands of spaghetti are intact, but the}

rest have each been snapped into two pieces. Of all the pieces of spaghetti from the rest have each been snapped into two pieces. Of all the pieces of spaghetti from the packet (broken and whole), what is the largest fraction guaranteed to be at least as packet (broken and whole), what is the largest fraction guaranteed to be at least as long as half an unbroken strand?

long as half an unbroken strand? (A) (A) 22 5 5 (B)(B) 3 3 5 5 (C)(C) 2 2 3 3 (D)(D) 1 1 2 2 (E)(E) 1 1 3 3 20.

20._{Mary has four children of diﬀerent ages, all under 10, and the product of their ages}_{Mary has four children of diﬀerent ages, all under 10, and the product of their ages}

is 2016. What is the sum of their ages? is 2016. What is the sum of their ages?

((AA) ) 330 0 ((BB) ) 334 4 ((CC) ) 228 8 ((DD) ) 229 9 ((EE) ) 3322

21.

21. _Ang_{Angelo has a 50}_{elo has a 50 L barrel of wat}_{L barrel of water and two si}_{er and two sizes of jug to}_{zes of jug to}

ﬁll

ﬁll, , larlarge and ge and smasmall. ll. EacEach jug, h jug, when fulwhen full, holds a l, holds a wholwholee number of litres.

number of litres.

He fills three large jugs, but does not have enough to fill He fills three large jugs, but does not have enough to fill a fourth

a fourth. . WitWith the water remah the water remainiining he then fills threeng he then fills three small jugs, but does not have enough to fill a fourth. small jugs, but does not have enough to fill a fourth. In litres, what is the capacity of the small jug?

In litres, what is the capacity of the small jug?

((AA) ) 5 5 ((BB) ) 4 4 ((CC) ) 3 3 ((DD) ) 2 2 ((EE) ) 11

22.

22. _{How many}_{How many 5-digit numbers contain all the digits 1, 2, 3, 4 and 5 and have the property}_{5-digit numbers contain all the digits 1, 2, 3, 4 and 5 and have the property}

that the diﬀerence between each pair of adjacent digits is at least 2? that the diﬀerence between each pair of adjacent digits is at least 2?

(20)

23.

23._{A number of people are standing in a line in such a way that each person is standing}_{A number of people are standing in a line in such a way that each person is standing}

nex

next t to exactlto exactly y one person who one person who is wearis wearing a ing a hat. hat. WhiWhich of ch of the follothe followinwing g coulcouldd not not

be the number of people standing in the line? be the number of people standing in the line?

((AA) ) 998 8 ((BB) ) 999 9 ((CC) ) 11000 0 ((DD) ) 11001 1 ((EE) ) 110022

24.

24._{Josh, Ruth and Sam each begin with a pile of lollies. From his pile Josh gives Ruth}_{Josh, Ruth and Sam each begin with a pile of lollies. From his pile Josh gives Ruth}

and Sam as many as each began with. From her new pile, Ruth gives Josh and Sam and Sam as many as each began with. From her new pile, Ruth gives Josh and Sam as many lollies as each of them then has. Finally, Sam gives Josh and Ruth as many as many lollies as each of them then has. Finally, Sam gives Josh and Ruth as many lollies as each then has.

lollies as each then has.

If in the end each has 32 lollies, how many did Josh have at the beginning? If in the end each has 32 lollies, how many did Josh have at the beginning?

((AA) ) 664 4 ((BB) ) 996 6 ((CC) ) 228 8 ((DD) ) 116 6 ((EE) ) 5522

25.

25._{A poem can have any number of lines and each line may rhyme with any of the other}_{A poem can have any number of lines and each line may rhyme with any of the other}

lines. lines.

For poems with only two lines, there are two diﬀerent rhyming structures: either the For poems with only two lines, there are two diﬀerent rhyming structures: either the lines rhyme or they do not.

lines rhyme or they do not. F

For or poems poems witwith h threthree e linlines, es, therthere e are are fivfive e diffdifferenerent t rhrhymiyming ng strustructurctures: es: eiteither allher all three lines rhyme, exactly one pair of lines rhyme (occurring in three ways), or none three lines rhyme, exactly one pair of lines rhyme (occurring in three ways), or none of the lines rhyme.

of the lines rhyme.

How many diﬀerent rhyming structures are there for poems with four lines? How many diﬀerent rhyming structures are there for poems with four lines?

((AA) ) 118 8 ((BB) ) 115 5 ((CC) ) 112 2 ((DD) ) 220 0 ((EE) ) 2266

26.

26._Digits_Digits aa_,, bb _and_and cc_{can be chose}_{can be}_{chosen to}_{n to mak}_{make the}_{e the follo}_{following multi}_{wing multiplicati}_{plication work.}_{on work. What is}_{What is}

the 3-digit number the 3-digit number abcabc_??

a a b b cc × × 2 2 44 11 c c b b aa ₂₂ 27.

27._{You have an unlimited supply of five different coloured}_{You have an unlimited supply of five different coloured}

pop-sticks, and want to make as many diﬀerent coloured pop-sticks, and want to make as many diﬀerent coloured equilateral triangles as possible, using three sticks. equilateral triangles as possible, using three sticks. One example is shown here.

One example is shown here.

Two triangles are not considered different if they are Two triangles are not considered different if they are rotations or reflections of each other.

rotations or reﬂections of each other.

How many diﬀerent triangles are possible?

How many diﬀerent triangles are possible? _blue_blue

r r e e d d r r e e d d 28.

28. _{What is}_{What is the larges}_{the largest}_{t 3-d}_3-digi_igit_{t nu}_{number that has}_{mber that has all of}_{all of its digi}_{its digits diﬀere}_{ts diﬀerent and}_{nt and is equal to}_{is equal to}

37 times the sum of its digits? 37 times the sum of its digits?

(21)

18

29.

29._{Lucas invented the list of numbers 2, 1, 3, 4, 7, ... where each number after the ﬁrst}_{Lucas invented the list of numbers 2, 1, 3, 4, 7, ... where each number after the ﬁrst}

tw

two is o is the sum of the sum of the previothe previous two. us two. He workHe worked out the ed out the ﬁrst 100 numﬁrst 100 numbers by hand,bers by hand, but unfortunately he made one mistake in the 90th number, which was out by 1. but unfortunately he made one mistake in the 90th number, which was out by 1. How far out was the 100th number?

How far out was the 100th number?

30.

30._T_To_{o mat}_match_{ch my}_{my hex}_hexag_agona_onally_{lly pa}_pave_ved_{d pa}_path,_{th, II}

bui

built lt aa Giant Giant’s ’s CauseCauseway way garden garden featfeatureure from 19 hexagonal stone columns, arranged from 19 hexagonal stone columns, arranged in a hexagonal pattern with three diﬀerent in a hexagonal pattern with three diﬀerent levels, as shown.

levels, as shown.

In how many ways can I climb from

In how many ways can I climb from S S _to_to F F

if I only step to an adjacent column, never if I only step to an adjacent column, never step on any column twice and never step step on any column twice and never step down a level? down a level? S S F F

(22)

2016 AMC – Intermediate Questions

2016 AMC – Intermediate Questions 19191919

Questions – Intermediate Division

1.

1. What iWhat is the vs the valualue of 20e of 20××16?16?

((AA) ) 33220 0 ((BB) ) 11440 0 ((CC) ) 2200116 6 ((DD) ) 332 2 ((EE) ) 880000

2.

2. In the ﬁgureIn the ﬁgure, the shaded r, the shaded region is wegion is what fractihat fraction of theon of the circle? circle? (A) (A) ₂₀₂₀11 (B)(B) ₁₀₁₀11 (C)(C) 11₂₂ (D) (D) ₆₀₆₀11 (E)(E) ₄₀₄₀11 1 1 2 2 1 1 4 4 1 1 5 5 3.

3. The cycliThe cycling road race throung road race through the Adelaigh the Adelaide Hillde Hills started at 11:1s started at 11:155 am and the winnam and the winnerer ﬁnished at 2:09

ﬁnished at 2:09 pm the same daypm the same day. . The winner’s time in minThe winner’s time in minutes wasutes was

((AA) ) 11335 5 ((BB) ) 11774 4 ((CC) ) 11664 4 ((DD) ) 22994 4 ((EE) ) 118866

4.

4. The The fractionfraction 720163 720163₂₀₁₆₂₀₁₆ isis

((AA) ) bebettwweeeen n 0 0 aannd d 1 1 ((BB) ) bebettwweeeen n 1 1 aannd d 110 0 ((CC) ) bbeettwweeeen n 110 0 aannd d 110000 ((DD) ) bbeettwweeeen n 11000 0 aannd d 1100000 0 ((EE) ) ggrreeaatteer r tthhaan n 11000000

5.

5. What iWhat is the vs the valualue of (1e of (1÷÷2)2)÷÷(3(3÷÷4)4) ??

(A)

(A) 22₃₃ (B)(B) 33₂₂ (C)(C) ₈3₈3 (D)(D) 11₆₆ (E)(E) ₂₄₂₄11

6.

6. 0.70.75% of a nu5% of a number imber is 6. s 6. The nThe numumber isber is

((AA) ) 88000 0 ((BB) ) 33000 0 ((CC) ) 1122000 0 ((DD) ) 44000 0 ((EE) ) 110000

7.

7. In the exIn the expressiopression below, n below, the letterthe letterss A A,, B,C,DB,C,D andand E E represent the numbers 1 represent the numbers 1,,22,,33,,44 and 5 in some order.

and 5 in some order.

A

A××BB ++ C C ××DD ++E E

What is the largest possible value of the expression? What is the largest possible value of the expression?

(23)

2016 AMC –

2016 AMC – Intermediate QuestionsIntermediate Questions

20

8.

8. _{In each of thes}_{In each of these square}_{e squares, the mark}_{s, the marked lengt}_{ed length is 1}_{h is 1 unit}_{unit. . Whi}_{Which of the squa}_{ch of the squares wo}_{res would}_uld

have the greatest perimeter? have the greatest perimeter?

P P.. 11 Q. Q. 11 R.R. 11 S. S. 11

((AA) ) P P ((BB) ) Q Q ((CC) ) R R ((DD) ) S S ((EE) ) aalll l aarre e tthhe e ssaammee

9.

9. _{On a clock}_{On a clock face, a lin}_{face, a line is dra}_{e is drawn betw}_{wn between 9 and 3 an}_{een 9 and 3 and another}_{d another}

betw

between 12 and een 12 and 8. 8. What is the acute anglWhat is the acute angle be betetweween theseen these lines? lines? (A) 45 (A) 45◦◦ (B) 60(B) 60◦◦ (C) 50(C) 50◦◦ (D) 30 (D) 30◦◦ _{(E) 22}_{(E) 22}_..₅₅◦◦ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 1212 10.

10. _The_{There are 3}_{re are 3 bl}_{blue pens, 4}_{ue pens, 4 re}_red_{d pens and 5}_{pens and 5 ye}_yell_{llow pens in a}_{ow pens in a box}_{box. . Wi}_Witho_{thout look}_{ut lookin}_{ing, I}_{g, I}

take pens from the box one by one. How many pens do I need to take from the box take pens from the box one by one. How many pens do I need to take from the box to be certain that I have at least one pen of each colour?

to be certain that I have at least one pen of each colour?

((AA) ) 8 8 ((BB) ) 9 9 ((CC) ) 110 0 ((DD) ) 111 1 ((EE) ) 1122

11.

11._{In the diagram, the value of}_{In the diagram, the value of}xx _is_is

((AA) ) 11220 0 ((BB) ) 11008 8 ((CC) ) 110055 ((DD) ) 11335 5 ((EE) ) 111122..₅₅ x x◦◦ x x◦◦ x x◦◦ x x◦◦ 12.

12._{How far is it from}_{How far is it from} AA_to_to BB_{measured in a straight}_{measured in a straight}

line? line? ((AA) ) 220 0 ((BB) ) 228 8 ((CC) ) 110 0 + + 99√ √ 22 (D) 8 + 9 (D) 8 + 9√ √ 2 2 ((EE) ) 1166 44 33 44 44 55 55 33 A A B B

(24)

2016 AMC – Intermediate Questions

2016 AMC – Intermediate Questions 21212121

13.

13._{A circle of radius 1 metre is inscribed inside a semicircle of}_{A circle of radius 1 metre is inscribed inside a semicircle of}

radi

radius 2 us 2 metrmetres. es. What is the area in What is the area in squsquare metreare metres of s of thethe semicircle

semicircle not not _{covered by the circle?}_{covered by the circle?}

(A) 2 (A) 2ππ _(B)_(B) ππ − − 11 ((CC) ) 22 (D) 2 (D) 2ππ − − 11 ((EE)) ππ 14.

14._{The value of}_{The value of}nn_{for which 4}_{for which 4}nn+1+1

=

= 221010 _is_is

((AA) ) 9 9 ((BB) ) 8 8 ((CC) ) 4 4 ((DD) ) 110 0 ((EE) ) 22

15.

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 3 3 ((EE) ) 44

16.

16._{What is the smallest number}_{What is the smallest number} N N _{for which}_{for which} 2016 2016

N

N is is a perfeca perfect sqt square?uare?

((AA) ) 114 4 ((BB) ) 2 2 ((CC) ) 556 6 ((DD) ) 112 2 ((EE) ) 77

17.

17._{Five people are sitting around a circle. Some always tell the truth, whilst the others}_{Five people are sitting around a circle. Some always tell the truth, whilst the others}

alw

alwayays s lielie. . EacEach h person claiperson claims to ms to be be sitsittinting g betwbetween two liareen two liars. s. HoHow w manmany y of themof them are telling the truth?

are telling the truth?

((AA) ) 0 0 ((BB) ) 1 1 ((CC) ) 2 2 ((DD) ) 3 3 ((EE) ) 44

18.

18. _{A cylin}_{A cylindric}_{drical glass of (insi}_{al glass of (inside) diam}_{de) diameter 6}_{eter 6 cm and heigh}_{cm and height 11}_{t 11 cm}_cm

is ﬁlled and then tilted to a 45

is ﬁlled and then tilted to a 45◦◦ angle so that some waterangle so that some water

overﬂows. How much water is left in it? overﬂows. How much water is left in it? (A) 48

(A) 48ππ_m_mL_L _((B_B)_{) 4455}ππ_m_mL_L _((C_C)_{) 6666}ππ_mL_mL

(D) 72

(D) 72ππ_m_mL_L _((E_E)_{) 6633}ππ_mL_mL

19.

19. _T_{Ten students sit a test consisting of}_{en students sit a test consisting of 20 questions.}_{20 questions. Two students}_{Two students get 8 questions correct}_{get 8 questions correct}

and one

and one stustudendent t gets 9 gets 9 quequestistions correcons correct. t. The remaiThe remaining sevning seven studenen students all ts all get atget at least 10 questions correct and the average number of questions answered correctly least 10 questions correct and the average number of questions answered correctly by these seven studen

by these seven students is an ts is an inteinteger. ger. If the average numIf the average number of ber of questiquestions answeredons answered correctly by all ten students is also an integer, then that integer is

correctly by all ten students is also an integer, then that integer is

(25)

2016 AMC –

2016 AMC – Intermediate QuestionsIntermediate Questions

22

20.

20._{This pedal-powered water pump is made}_{This pedal-powered water pump is made}

from bicycle parts. from bicycle parts.

A 30-tooth gear on the pedals has a chain A 30-tooth gear on the pedals has a chain to a 15-tooth gear. On the same axle as the to a 15-tooth gear. On the same axle as the 15-tooth gear is a 32-tooth gear that drives 15-tooth gear is a 32-tooth gear that drives a chain to a 40-tooth gear on the pump. a chain to a 40-tooth gear on the pump. For every 100 complete revolutions of the For every 100 complete revolutions of the pedals, how many times does the gear on pedals, how many times does the gear on the pump turn?

the pump turn?

((AA) ) 11660 0 ((BB) ) 22550 0 ((CC) ) 110077 ((DD) ) 993 3 ((EE) ) 337711 2 2 P P U U M M P P 30 30 3232 15 15 40 40 21.

21. _{A gardene}_{A gardener}_{r wis}_{wishes to}_{hes to put a}_{put a circ}_{circular wat}_{ular water feature (pool) in}_{er feature (pool) in}

a right

a right-ang-angled trialed triangulngular plot that has sides of 6ar plot that has sides of 6 m and m and 88 mm on its two sma

on its two smallellest sidesst sides. . WhaWhat t is the radius in metreis the radius in metres of s of the largest pool that will ﬁt?

the largest pool that will ﬁt? (A) 2

(A) 2

√

22

₋

1 1 ((BB) ) 2 2 ((CC) ) 4 4 ((DD) ) 3 3 ((EE) ) 22

√

22

22.

22._{A sequence of 10 letters is made according to the following rules.}_{A sequence of 10 letters is made according to the following rules.}

••

The letter P can only be followed by Q or R. The letter P can only be followed by Q or R.

••

The letter Q can only be followed by R or S. The letter Q can only be followed by R or S.

••

The letter R can only be followed by S or T. The letter R can only be followed by S or T.

••

The letter S can only be followed by T or P. The letter S can only be followed by T or P.

••

The letter T can only be followed by P or Q. The letter T can only be followed by P or Q.

How many possible sequences are there where the ﬁrst, fourth, and tenth letters are How many possible sequences are there where the ﬁrst, fourth, and tenth letters are all Q?

all Q?

((AA) ) 663 3 ((BB) ) 339 9 ((CC) ) 332 2 ((DD) ) 445 5 ((EE) ) 3366

23.

23. _Cyn_{Cynthia’s afternoon train normall}_{thia’s afternoon train normally arrives at her station at}_{y arrives at her station at 5:30}_{5:30 pm each day}_{pm each day, where}_{, where}

she is picked up by Alan and driven home. she is picked up by Alan and driven home. One day she was on an earlier train whic

One day she was on an earlier train which arrived at 5h arrived at 5 pm, and she decided to walk inpm, and she decided to walk in the directio

the direction Alan was cominn Alan was coming from home. g from home. Alan had left in time to meet the 5:30Alan had left in time to meet the 5:30 pmpm train, but this time he picked up Cynthia and they arrived home 10 minutes earlier train, but this time he picked up Cynthia and they arrived home 10 minutes earlier than usual.

than usual.

For how many minutes had Cynthia walked before Alan picked her up? For how many minutes had Cynthia walked before Alan picked her up?