2017 澳大利亚数学AMC11-12(真题）

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?

(A) (B) (C) (D) (E)

2. What is the value of 17a + 20b when a = 20 and b = 17?

(A) 680 (B) 689 (C) 1720 (D) 2017 (E) 3737

3. 1000% of 1 is

(A) 0.1 (B) 1 (C) 10 (D) 100 (E) 1000

4. 42 + 33 + 24 =

(A) 29 (B) 33 (C) 43 (D) 59 (E) 73

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?

(A) 30° (B) 40° (C) 45° (D) 50° (E) 60°

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?

(A) 3 (B) 4 (C) 6 (D) 7 (E) 8

7. 200 − 199 + 198 − 197 + 196 − ··· + 2 − 1 is equal to

(A) 1 (B) 99 (C) 100 (D) 101 (E) 200

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?

(A) 54 (B) 55 (C) 56 (D) 57 (E) 58

9. I make a multiplication table for the numbers from 1 to 5. What is the sum of the 25 numbers inside this table?

(A) 100 (B) 200 (C) 225 (D) 250 (E) 256

10. Evaluate .

(A) 2 (B) 4 (C) 8 (D) 80 (E) 800

11. Each row, column and diagonal of this grid should add up to 30.

What number goes in the bottom-right corner?

(A) 3 (B) 15 (C) 5 (D) 13 (E) 4

12. The vertices of the innermost white square bisect the sides of the shaded square. What is the area of the innermost white square?

(A) 6 (B) 4 (C) 10 (D) 3 (E) 5

13. Two cars leaving Sale for Melbourne are travelling at 60 km/h and are 240 m apart travelling west on the Princes Highway. When each car reaches the 80 km/h sign it starts travelling at 80 km/h. How far apart are they once they are both past the sign?

(A) 180 m (B) 240 m (C) 360 m (D) 300 m (E) 320 m

14. In the triangle shown, PQ = SQ = SR = QR and ∠STR = 90◦.

The ratio TR : PS is equal to

(A) (B) (C) 1 : 2 (D) (E)

15. The solution to the equation 2017x2 − 20172017 = 0 is

(A) x = ±2017 (B) x = ±20172017 (C) x = ±20171008

(D) x = ±20171009 (E)

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

(A) (B) 30° (C) 45° (D) 60° (E)

17. I spent exactly $4000 on books which cost $25 and $26 each. I bought at least one book at each price. What is the maximum number of books I could have bought?

(A) 153 (B) 157 (C) 155 (D) 159 (E) 161

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?

(A) 1 (B) 9 (C) 99 (D) 247 (E) 315

19. Two circles of diameter 12 cm overlap, forming the shape shown with perimeter 14π cm. What is the area of the shape, in cm2?

(A) 42π + 18 (B) (C) 72π − 18

(D) (E)

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?

(A) 39 cm (B) 42 cm (C) 45 cm (D) 48 cm (E) 50 cm

21. A square is drawn in the corner of a right-angled triangle with side lengths a, b and c, as shown.

Which expression gives the ratio of the unshaded area to the shaded area in all cases?

(A) 1:1 (B) c : (a + b) (C) ab : c2 (D) (a + b)2 : 2c2 (E) c2 : 2ab

22. Andy and Brandy have decided to share a cake in an unusual way. Andy will take some fraction of the cake first. From then on, Brandy and Andy will alternate taking half of what remains each time. Eventually there is so little left they decide they are finished. What fraction of the cake should Andy take first to ensure that they end up with half each?

(A) (B) (C) (D) (E)

23. Each student in a class sends ten Christmas cards, one to each of ten other students in the class. It is known that no two students sent Christmas cards to each other.

What is the smallest number of students who could be in the class?

(A) 15 (B) 20 (C) 21 (D) 25 (E) 30

24. A circle touches one side of a square and passes through the end vertices of the opposite side.

The circle has circumference c and the square has perimeter s. Which of the following is true to the nearest one percent?

(A) c is approximately 5% larger than s.

(B) c is approximately 2% larger than s.

(D) c is approximately 2% smaller than s.

(E) c is approximately 5% smaller than s.

25. I have two blue socks, two red socks, and two yellow socks in a drawer. On Monday, I randomly pick two of the socks to wear. On Tuesday, I randomly pick two of the remaining socks to wear. On Wednesday, I wear the two remaining socks from the drawer.

What is the probability that I didn’t wear a matching pair of socks on any of the

three days?

(A) (B) (C) (D) (E)

26. A river runs from west to east. Colin’s hat has fallen in 28 metres upstream from the waterfall. Pepper the wonder dog is admiring the view 36 metres directly south of the waterfall. Pepper can run exactly three times as fast as the hat is being carried downstream. What is the minimum distance in metres that Colin’s hat must travel before Pepper can retrieve it?

27. Mike multiplied at least two consecutive integers together. He obtained a six-digit number N. The first two digits of N are 47 and the last two digits of N are 74. What is the sum of the integers that Mike multiplied together?

28. For n ⩾ 3, the sequence of centred n-gon numbers is found by starting with a central dot, then adding layers consisting of n-gons of dots around this centre, where the number of dots on each side increases by 1 for each layer.

For instance, the sequence of centred 7-gon numbers starts 1, 8, 22, 43,... as shown.

What is the smallest n for which 2017 is in the sequence of centred n-gon numbers?

29. A friend and I play a game. We each start with two coins. We take it in turns to toss a coin; if it comes down heads, we keep it, if tails, we give it to the other. I always go first, and the game ends when one of us wins by having all four coins.

If we play this game 840 times, what is the expected number of games that I would win?

30. The squares of an 8×8 grid are coloured black or white. A colouring is called balanced if each 2 × 2 subgrid contains exactly two squares of each colour. An example of a balanced colouring of a 3 × 3 grid is shown on the left. The unbalanced colouring on the right fails this requirement since the 2 × 2 subgrid on the bottom right contains three white squares.