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2018 AMC Intermediate Questions Questions { Intermediate Division 1. The value of 2018 18 1000 is (A) 0.02 (B) 0.1 (C) 1 (D) 2 (E) 2000 2. What value is indicated on this charisma-meter? (A) 36.65 (B) 37.65 (C) 38.65 (D) 37.15 (E) 37.3 38 36 3. What is the di erence between the sum and the product of 4 and 5? (A) 1 (B) 8 (C) 9(D) 11 (E) 20 4. 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  5. Which of the following is not a whole number? (A) 350 2 (B) 350 7 (C) 350 5 (D) 350 25 (E) 350 20 6. Nora, Anne, Warren and Andrew bought plastic capital letters to spell each of their names on their birthday cakes. Their birthdays are on di erent dates, so they planned to reuse letters on di erent cakes. What is the smallest number of letters they needed? (A) 8 (B) 9(C) 10 (D) 11 (E) 12 A A A N N N N N N E E E 7. In y ears, 2018 days is closest to (A) 4.5 years (B) 5 years (C) 5.5 years (D) 6 years (E) 6.5 years c Australian Mathematics Trust www.amt.edu.au 22

2018 AMC Intermediate Questions 8. Two paths from Ato Care pictured. The stepped path consists of horizontal and vertical segments, whereas the dashed path is straight. What is the di erence in length between the two paths? (A) 1 m (B) 2 m(C) 3 m (D) 4 m (E) 0 m 4 m 3 m A B C 9. The value of 9 1:2345 9 0:1234 is (A) 9.9999 (B) 9 (C) 9.0909 (D) 10.909 (E) 11.1111 10. What fraction of this regular hexagon is shaded? (A)1 2 (B) 2 3 (C) 3 4 (D) 3 5 (E) 4 5 11. 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 12. In a certain year there were exactly four Tuesdays and exactly four Fridays in the month of December. What day of the week was 31 December? (A) Monday (B) Wednesday (C) Thursday (D) Friday (E) Saturday 13. Fill in this diagram so that each of the rows, columns and diago- nals adds to 18. What is the sum of all the corner numbers? (A) 20 (B) 22 (C) 23 (D) 24 (E) 25 4 6 14. The sum of 4 consecutive integers is t. In terms of t, the smallest of the four integers is (A) t 10 4 (B) t 2 4 (C) t 3 4 (D) t 4 4 (E) t 6 4 c Australian Mathematics Trust www.amt.edu.au 23

2018 AMC Intermediate Questions 15. A 3-dimensional ob ject is formed by gluing six identical cubes together. Four of the diagrams below show this ob ject viewed from di erent angles, but one diagram shows a di erent ob ject. Which diagram shows the di erent ob ject? (A) (B) (C) (D) (E) 16. 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 17. Allan and Zarwa are playing a game tossing a coin. Allan wins as soon as a head is tossed and Zarwa wins if two tails are tossed. The probability that Allan wins is (A)1 2 (B) 3 5 (C) 5 8 (D) 2 3 (E) 3 4 18. In this expression 1 3 1 4 1 5 1 6 1 7 we place either a plus sign or a minus sign in each box so that the result is the smallest positive number possible. The result is (A) between 0 and 1 100 (C) between 1 50 and 1 20 (B) between 1 100 and 1 50 (D) between 1 20 and 1 10 (E) between 1 10 and 1 c Australian Mathematics Trust www.amt.edu.au 24

2018 AMC Intermediate Questions 19. A town is laid out in a square of side 1 kilometre, with six straight roads as shown. Each day the postman must walk the full length of every road at least once, starting wherever he likes and ending wherever he likes. How long is the shortest route he can take, in kilometres? (A) 4 + p 2 2 (B) 4 +p 2 (C) 4 + 2 p 2 (D) 4 + 3 p 2 (E) 5 + 2 p 2 20. A rectangle with integer sides has a diagonal stripe which starts 1 unit from the diagonal corners, as in the diagram. The area of the stripe is exactly half of the area of the rectangle. What is the perimeter of this rectangle? (A) 14 (B) 16 (C) 18 (D) 20 (E) 22 1 1 1 1 21. How many digits does the number 20 18 have? (A) 24 (B) 38 (C) 18(D) 36 (E) 25 22. 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 23. Suppose pis a two-digit number and qhas the same digits, but in reverse order. The number p2 q2 is a non-zero perfect square. The sum of the digits of pis (A) 7 (B) 9(C) 11 (D) 12 (E) 13 c Australian Mathematics Trust www.amt.edu.au 25

2018 AMC Intermediate Questions 24. In triangle 4P QR,U is a point on P R,S is a point on P Q,Tis a point on QRwith U S kRQ, and U TkP Q. The area of 4P S Uis 120 cm 2 and the area of 4T U R is 270 cm 2 . The area of 4QS T, in square centimetres, is (A) 150 (B) 160 (C) 170 (D) 180 (E) 200< >> > > < P Q R S T U 120 270 25. This year Ann's age is the sum of the digits of her maths teacher's age. In ve years Ann's age will be the product of the digits of her maths teacher's age at that time. How old is Ann now? (A) 11 (B) 13(C) 15 (D) 14 (E) 16 26. I have a three-digit number, and I add its digits to create its digit sum. When the digit sum of my number is subtracted from my number, the result is the square of the digit sum. What is my three-digit number? 27. A road from Tamworth to Broken Hill is 999 km long. There are road signs each kilometre along the road that show the distances (in kilometres) to both towns as shown in the diagram. 0j999 1j998 2j997 3j996


998j1 999j0 How many road signs are there that use exactly two di erent digits? 28. In the division shown, X,Y and Zare di erent non-zero digits. 8 X Y Z Z X rem.Y What is the three-digit number X Y Z? 29. An in nite increasing list of numbers has the property that the median of the rst nterms equals the nth odd positive integer. How many numbers in the list are less than 2018? c Australian Mathematics Trust www.amt.edu.au 26

2018 AMC Intermediate Questions 30. 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? c Australian Mathematics Trust www.amt.edu.au 27