DISINI j2s2005

THE HARMONY SOUTH AFRICAN

MATHEMATICS OLYMPIAD

Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION Sponsored by HARMONY GOLD MINING

SECOND ROUND 2005

JUNIOR SECTION: GRADES 8 AND 9 10 MAY 2005

TIME: 120 MINUTES NUMBER OF QUESTIONS: 20

ANSWERS PRACTICE

EXAMPLES

POSITION

1 C

2 D

NUMBER POSITION 1 C 2 D 3 E 4 C 5 A 6 C 7 D 8 C 9 C 10 E 11 C & D

12 A 13 A 14 B 15 B 16 C 17 E 18 B 19 D 20 E

PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 E-mail: ellie@samf.ac.za

Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns

1. ANSWER: C

4 3

4 1 1

2 1 2 1 1

= − =

× −

2. ANSWER: D

9 x 7

8 10 a

b c d

The sum of the rows, columns and diagonals must be the same.

Row1 = Diagonal from left

∴

+ + = + + ∴ = − 9 x 7 9 10 d

d x 3

Also Row 1 = Column 1

∴ + + = + + = − 9 x 7 9 8 6

b x 1

Lastly Row 3 = Column 2

∴ − + + − = + + ∴ =

x 1 c x 3 x b c x 14

3. ANSWER: E

Suppose the initial purchase price is X.

Applying a 20% discount will lead to Terry having to pay X

X

X −0,2 =0,8 .

Applying a further 10% discount will lead to a final price of X

X

X 0,8 0,72 8

,

0 − = .

The effective single discount is therefore

% 28 28

, 0 72 ,

0 = =

− X X

4. ANSWER: C

T U T U x y y x

∴ 10x + y + 10y + x =11x+11y

(

)

(

x y

)

y x

+ × =

+ =

11 11

5. ANSWER: A

Let the 2 numbers be x and y.

7. y x and

8 y x Then

= ×

− = +

Split 7 into 2 factors: 1×7= x×y.

Since the sum = –8,

then the numbers are: –1 and –7.

6. ANSWER: C

36 , 1 32 , 3 32 ,

4 2 − 2 +

Let us investigate

5 4 9

2 32 2

= − =

−

9 16 25

4 52 2

= − =

−

2 2

3 −2 = + =3 2 5 ; 52−42 =5+4

Conclusion: a2−b2 =a+b if a−b=1 Therefore: 4,322−3,322+1,36

9

1,36 7,64

1,36 3,32

4,32

=

+ =

+ +

7. ANSWER: D etc. numerator, s bracket' next the cancels r denominato first the as 2005 2 2005 2004 80 79 79 78 5 4 4 3 3 2 2005 1 1 5 1 1 4 1 1 3 1 1 =                               =       ₋       ₋       ₋       ₋ L L L

8. ANSWER: C

hours 2000 10 2 , 0 10 6 10 2 , 1 Speed Distance Time 4 4 8 = × = × × = =

The answer is just less than 3 months.

hours 2232 approx. months 3 hours 1488 approx. months 2 hours 744 approx. 24 31 approx. month 1 = = = × =

(

)

hours 744 24 720 24 1 30 24 31 = + = × + = ×

9. ANSWER: C

D S T

Speed =

Time Distance

In quadrilateral ADHE 0 0 1 0 0 1 360 90 ˆ 90 ˆ 360 ˆ ˆ ˆ ˆ = + + + = + + + H A H E A H H D A A A C H B C H B H H A − = = + = = + 0 0 1 0 1 180 ˆ 180 C Hˆ B Aˆ angles) opposite y (verticall ˆ ˆ 180 ˆ ˆ OR

(

)

A A H A C ADC ˆ 180 ˆ 90 90 ˆ angle external ˆ 90 ˆ , In 0 0 0 2 0 − = − + = ∴ − = ∆

10. ANSWER: E

Smallest possible y x

-values can be:

20 2 1 10 or 5 2 10 or 12 2 1 6 or 3 2

6 ₌₋

− − = − = − − = − −

Similarly the biggest value for b y x = =12 240 12 20 12 and 20 − = × − = ∴ = − = ∴ ab b a

11. ANSWER: C & D

Maximum number of days in April is 30. When April starts on a Saturday the calendar will look as follows. The 28th of April 20xy will fall on a Friday. Sun-day Mon-day Tues-day Wed-nesday Thurs-day Friday Satur-day 1

3 7

10 14

17 21

24 28

30

However, when April starts on a Tuesday the 28th of April 20xy will fall on a Monday.

12. ANSWER: A

Area of each face = 1 cm2

2

Number of exposed faces 5 4 4 4 5 100 terms 98 4 2 5

392 10 402 Area of exposed faces 402cm

= + + + + + = × + × = + = =

L L

13. ANSWER: A

0

0

0 0 0

0 0

0

0

ˆ ˆ

Let

ˆ ˆ ₄₀

ˆ ˆ ˆ ˆ ₃₆₀

40 40 360

4 80 360

4 280

70

A x C

B D x

A B C D

x x x x

x x x

= = = = + + + + =

+ + + + + = + = = =

14. ANSWER: B

total earnings average

days worked 78

20 1560

Also 90 25 2250

x x

y y

= ∴ = ∴ = = ∴ =

She must earn R2 250 – R1 560 = R690 over the next 5 days.

15. ANSWER: B

A = B + x B

(

)

(

)

(

)

(

)

(

)

(

)

10

7 rem

10 10

7

2 2

2

10 10 7 2

10 10 14 7

11 10 14 8

2 3

2 3

3 2

if 2 then 3, and 5.

if 4 then 6, and 10; too big 3.

B x B

x

B x B

B x B x

B x B x

LCM B x

B x B B x x

B x B B x x

B x B x

x B

x B B x

B x A

B x A x

+ + = + +

+ _{+ = +}

+ +

= +

∴ + + = + +

∴ + + = + +

∴ + = +

∴ = ∴ = ∴ =

= = =

16. ANSWER: C

The number can be written as

(

100×2

) (

+ 10×y

) (

+ 2x−1

)

3 30 10 10

29 112 1 10 102

1 2 10 100

+ =

+ =

+ =

− + =

− + + =

x y

x y

x y

x

x y x

17. ANSWER: E

Trial and improvement:

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

18. ANSWER: B

Let genuine = g and counterfeit = c Then 6 (3g + 3c)

(g g c) 3 3 (g c c) or

( ) ( )

₁₄₂₄₃ possible

) 1 ( ) 1 (

ccc ggg

lighter 3 (g c c)

(c) 1 1 (g) 1 (c) problem solved or (c) 1 1 (c) 1 (g) problem solved

2 x weighings (minimum) or separated into lots of 2

19. ANSWER: D

The smallest possible sum will be 1 (from 10 000 000) and the highest possible sum will be 72 (from 99 999 999).

We therefore have 72 possible answers and the average of 1 and 72 will be

(

)

2 1 36 72 1 2 1

= +

∴ Both 36 and 37 will occur the most.

20. ANSWER: E

2

2

2

2

2

1 (4)

Area DOE 4

4 4

1 (2)

Area DAC

4 4

1

Area CBE as before

4

cm cm cm

π _π

π _π

π

= =

= =

=

Area Area Area Area Area leaf

4 2 2 area leaf OC

DCEF DOE DCO OCE OC

π π π

= − − +

= − − +

(

)

(

)

(

)

Area DCEF Area leaf OC Then Area of leaf is 2 2

Answer: 2 2 2 4 2 4 8

π

π π π

∴ =

 − 

 

 − = − = −

7. ANSWER: D etc. numerator, s bracket' next the cancels r denominato first the as 2005 2 2005 2004 80 79 79 78 5 4 4 3 3 2 2005 1 1 5 1 1 4 1 1 3 1 1 =                               =       ₋       ₋       ₋       ₋ L L L

8. ANSWER: C

hours 2000 10 2 , 0 10 6 10 2 , 1 Speed Distance Time 4 4 8 = × = × × = =

The answer is just less than 3 months.

hours 2232 approx. months 3 hours 1488 approx. months 2 hours 744 approx. 24 31 approx. month 1 = = = × =

(

)

hours 744 24 720 24 1 30 24 31 = + = × + = ×

9. ANSWER: C

D S T

Speed = Time Distance

In quadrilateral ADHE 0 0 1 0 0 1 360 90 ˆ 90 ˆ 360 ˆ ˆ ˆ ˆ = + + + = + + + H A H E A H H D A A A C H B C H B H H A − = = + = = + 0 0 1 0 1 180 ˆ 180 C Hˆ B Aˆ angles) opposite y (verticall ˆ ˆ 180 ˆ ˆ OR

(

)

A A H A C ADC ˆ 180 ˆ 90 90 ˆ angle external ˆ 90 ˆ , In 0 0 0 2 0 − = − + = ∴ − = ∆

10. ANSWER: E Smallest possible

y x

-values can be:

20 2 1 10 or 5 2 10 or 12 2 1 6 or 3 2

6 ₌₋

− − = − = − − = − −

Similarly the biggest value for b y x = =12 240 12 20 12 and 20 − = × − = ∴ = − = ∴ ab b a

11. ANSWER: C & D

Maximum number of days in April is 30. When April starts on a Saturday the calendar will look as follows. The 28th of April 20xy will fall on a Friday. Sun-day Mon-day Tues-day Wed-nesday Thurs-day Friday Satur-day 1

3 7

10 14

17 21

24 28

30

However, when April starts on a Tuesday the 28th of April 20xy will fall on a Monday.

12. ANSWER: A

Area of each face = 1 cm2

2

Number of exposed faces 5 4 4 4 5 100 terms 98 4 2 5

392 10 402 Area of exposed faces 402cm

= + + + + + = × + ×

= +

= =

L L

13. ANSWER: A

0 0

0 0 0

0 0

0 0

ˆ ˆ

Let

ˆ ˆ ₄₀

ˆ ˆ ˆ ˆ ₃₆₀

40 40 360

4 80 360 4 280 70 A x C B D x A B C D

x x x x x

x x

= = = = + + + + =

+ + + + + =

+ =

= =

14. ANSWER: B total earnings average

days worked 78

20 1560

Also 90 25 2250

x x

y y

= ∴ = ∴ = = ∴ =

She must earn R2 250 – R1 560 = R690 over the next 5 days.

15. ANSWER: B

A = B + x B

(

)

(

)

(

)

(

)

(

)

(

)

10

7 rem 10 10

7

2 2

2

10 10 7 2

10 10 14 7

11 10 14 8 2 3 2 3

3 2

if 2 then 3, and 5.

if 4 then 6, and 10; too big 3. B x B

x B x B

B x B x

B x B x

LCM B x

B x B B x x

B x B B x x

B x B x

x B x B

B x

B x A

B x A x

+ + = + +

+ _{+ = +}

+ +

= +

∴ + + = + +

∴ + + = + +

∴ + = +

∴ =

∴ =

∴ =

= = =

16. ANSWER: C

The number can be written as

(

100×2

) (

+ 10×y

) (

+ 2x−1

)

3 30 10 10

29 112 1 10 102

1 2 10 100

+ =

+ =

+ =

− + =

− + + =

x y

x y

x y

x

x y x

17. ANSWER: E

Trial and improvement:

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

18. ANSWER: B

Let genuine = g and counterfeit = c Then 6 (3g + 3c)

(g g c) 3 3 (g c c) or

( ) ( )

₁₄₂₄₃ possible

) 1 ( ) 1 (

ccc ggg lighter

3 (g c c)

(c) 1 1 (g) 1 (c) problem solved or (c) 1 1 (c) 1 (g) problem solved

2 x weighings (minimum) or separated into lots of 2

19. ANSWER: D

The smallest possible sum will be 1 (from 10 000 000) and the highest possible sum will be 72 (from 99 999 999).

We therefore have 72 possible answers and the average of 1 and 72 will be

(

)

2 1 36 72 1 2 1

= +

∴ Both 36 and 37 will occur the most.

20. ANSWER: E

2

2

2

2

2

1 (4)

Area DOE 4

4 4

1 (2)

Area DAC

4 4

1

Area CBE as before 4

cm cm cm

π _π

π _π

π

= =

= =

=

Area Area Area Area Area leaf 4 2 2 area leaf OC

DCEF DOE DCO OCE OC

π π π

= − − +

= − − +

(

)

(

)

(

)

Area DCEF Area leaf OC Then Area of leaf is 2 2

Answer: 2 2 2 4 2 4 8

π

π π π

∴ =

 − 

 

 − = − = −