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2018 AMC Senior Questions Questions { Senior Division 1. In the diagram, P QRSis a square. What is the size of \X P Y? (A) 25  (B) 30 (C) 35 (D) 40  (E) 45 P Q RS X Y 25  25  2. The Gre at North Walk is a 250 km long trail from Sydney to Newcastle. If you want to complete it in 8 days, approximately how far do you need to walk each day? (A) 15 km (B) 20 km (C) 30 km (D) 40 km (E) 80 km 3. Half of a number is 32. What is twice the number? (A) 16 (B) 32(C) 64 (D) 128 (E) 256 4. What fraction of this regular hexagon is shaded? (A)1 2 (B) 2 3 (C) 3 4 (D) 3 5 (E) 4 5 5. The v alue of 9 1:2345 9 0:1234 is (A) 9.9999 (B) 9 (C) 9.0909 (D) 10.909 (E) 11.1111 6. What is 2 0 18 ? (A) 0 (B) 1(C) 2 (D) 3 (E) 10 7. 1000% of a number is 100. What is the number? (A) 0.1 (B) 1(C) 10 (D) 100 (E) 1000 8. The cost of feeding four dogs for three days is $60. Using the same food costs per dog per day, what would be the cost of feeding seven dogs for seven days? (A) $140 (B) $200 (C) $245 (D) $350 (E) $420 c Australian Mathematics Trust www.amt.edu.au 28

2018 AMC Senior Questions 9. In the triangle ABC,M is the midpoint of AB. Which one of the following statements must be true? (A) \C AM =\AC M (C) AC= 2BC (B) \C M B = 2\C AM (D) C M =B C (E) Area 4AM C= Area4M B C A BC M 10. The sum of the numbers from 1 to 100 is 5050. What is the sum of the numbers from 101 to 200? (A) 15 050 (B) 50 500 (C) 51 500 (D) 150 500 (E) 505 000 11. Leila has a number of identical equilateral triangle shaped tiles. How many of these must she put together in a row (edge to edge) to create a shape which has a perimeter ten times that of a single tile? (A) 14 (B) 20 (C) 25(D) 28 (E) 30 12. In the circle shown, Cis the centre and A,B,D and E all lie on the circumference. Re ex \B C D = 200 , \DC A =x and \B C A = 3x as shown. The ratio of \DAC:\B AC is (A) 3 : 1 (B) 5 : 2(C) 8 : 3 (D) 7 : 4 (E) 7 : 3 200  x  3x  A B C D E 13. Instead of multiplying a number by 4 and then subtracting 330, I accidentally divided that number by 4 and then added 330. Luckily, my nal answer was correct. What was the original number? (A) 220 (B) 990(C) 144 (D) 374 (E) 176 14. The diagram shows a regular octagon of side length 1 metre. In square metres, what is the area of the shaded region? (A) 1 (B)p 2 (C) 2 (D) 3 p 2 (E) 1 +p 2 2 c Australian Mathematics Trust www.amt.edu.au 29

2018 AMC Senior Questions 15. A netball coach is planning a train trip for players from her two netball clubs, Panthers and Warriors. The two clubs are in di erent towns, so the train fares per player are di erent. For the same cost she can either take 6 Panthers and 7 Warriors or she can take 8 Panthers and 4 Warriors. If she takes only members of the Warriors on the train journey, the number she could take for the same cost is (A) 11 (B) 13 (C) 16(D) 20 (E) 25 16. The triangle P QRshown has a right angle at P. Points Tand Sare the midpoints of the sides P R and P Q, respectively. Also \QT P= and \S RP = . The ratio tan : tan equals (A) 3 : 1 (B) 4 : 1(C) 5 : 1 (D) 7 : 2 (E) 9 : 2 P Q RS T k k j j 17. Three fair 6-sided dice are thrown. What is the probability that the three numbers rolled are three consecutive numbers, in some order? (A) 1 6 (B) 1 9 (C) 1 27 (D) 7 36 (E) 1 54 18. How many digits does the number 20 18 have? (A) 24 (B) 38 (C) 18(D) 36 (E) 25 19. In this subtraction, the rst number has 100 digits and the second number has 50 digits. 111: : : : : : 111 | {z } 100 digits 222 : : :222 | {z } 50 digits What is the sum of the digits in the result? (A) 375 (B) 420 (C) 429(D) 450 (E) 475 20. I have two regular polygons where the larger polygon has 5 sides more than the smaller polygon. The interior angles of the two polygons di er by 1  . How many sides does the larger polygon have? (A) 30 (B) 40(C) 45 (D) 50 (E) 60 c Australian Mathematics Trust www.amt.edu.au 30

2018 AMC Senior Questions 21. How many solutions ( m; n) exist for the equation n= p 100 m2 where both mand nare integers? (A) 4 (B) 6 (C) 7 (D) 8 (E) 10 22. A tetrahedron is inscribed in a cube of side length 2 as shown. What is the volume of the tetrahedron? (A)8 3 (B) 4 (C) 16 3 (D) p 6 (E) 8 2p 2 23. A rectangle has sides of length 5 and 12 units. A diagonal is drawn and then the largest possible circle is drawn in each of the two triangles. What is the distance between the cen- tres of these two circles? (A) p 60 (B) 8 (C)p 65 (D) p 68 (E) 9 24. In the equation r q : : : p 256 | {z } 60 = 2 (8 x ) the value of xis (A) 17 (B)19 (C)21 (D)23 (E) 16 25. A right-angled triangle with sides of length 3, 4 and 5 is tiled by in nitely many right- angled triangles, as shown. What is the shaded area? (A)18 7 (B) 54 25 (C) 8 3 (D) 27 17 (E) 96 41 43 5 26. LetAbe a 2018-digit number which is divisible by 9. Let Bbe the sum of all digits of A and Cbe the sum of all digits of B. Find the sum of all possible values of C. c Australian Mathematics Trust www.amt.edu.au 31

2018 AMC Senior Questions 27. The trapezium ABC DhasAB = 100, BC= 130, C D = 150 and DA= 120, with right angles at Aand D . An interior point Qis joined to the midpoints of all 4 sides. The four quadrilaterals formed have equal areas. What is the length AQ? A B CD Q 28. Donald has a pair of blue shoes, a pair of red shoes, and a pair of white shoes. He wants to put these six shoes side by side in a row. However, Donald wants the left shoe of each pair to be somewhere to the left of the corresponding right shoe. How many ways are there to do this? 29. Forn 3, a pattern can be made by overlapping ncircles, each of circumference 1 unit, so that each circle passes through a central point and the resulting pattern has order- nrotational symmetry. For instance, the diagram shows the pattern where n= 7. If the total length of visible arcs is 60 units, what is n? 30. Consider an n ngrid lled with the numbers 1; : : : ; n 2 in ascending order from left to right, top to bottom. A shueconsists of the following two steps:  Shift every entry one position to the right. An entry at the end of a row moves to the beginning of the next row and the bottom-right entry moves to the top-left position.  Then shift every entry down one position. An entry at the bottom of a column moves to the top of the next column and again the bottom-right entry moves to the top-left position. An example for the 3 3 grid is shown. Note that the two steps shown constitute one shue. shift right shift down 1 2 3 4 5 6 7 8 9 9 1 2 3 4 5 6 7 8 8 6 7 9 1 2 3 4 5 What is the smallest value of nfor which the n ngrid requires more than 20 000 shues for the numbers to be returned to their original order? c Australian Mathematics Trust www.amt.edu.au 32