Preface Created in 1977 by Dr. George Lenchner, an internationally known math educator, the Math Olympiads went public in 1979. Last year nearly 170,000 students from 6,000 teams worldwide participated in the Olympiads. All 50 states and about 30 other countries were represented. In ASEAN, it is known as American Mathematics Olympiad (AMO), www.amo.sg and is organized by Singapore International Math Contests Centre (SIMCC). This is an excellent contest to introduce middle and upper primary students to the exciting world of mathematics competitions, without discouraging them with very tough questions. So, let them take part in easier competitions to build their confidence and interest, because the fastest way to kill interest is to fail badly. So, join AMO before taking on tougher math competitions like SASMO, NMOS, Mathlympics, RIWPMC and SMOPS. The past year contest paper with detailed solutions and comments provided will enhance your problem solving skills as well as your ability to think about and solve complex problems. We hope to expose more students from Primary 3 and up to a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. We can then see them thinking and applying concepts they have learnt as well as utilising a common sense approach to solve problems. The aims of the American Mathematics Olympiad are: o To stimulate enthusiasm and a love for Mathematics o To introduce important Mathematical concepts o To teach major strategies for problem solving o To develop Mathematical flexibility in solving problems o To strengthen Mathematical intuition o To foster Mathematical creativity and ingenuity o To provide for the satisfaction, joy, and thrill of meeting challenges We hope to introduce more students to Math Olympiad competitions in a more comfortable and interesting way that will also sustain their mathematical interest for the years to come and hopefully we will succeed in developing many Scholars from these competitions. CONTENTS Competition Details 1 AMO Past Year Problems 5 Solutions for AMO Past Year Problems 14 Competition Details American Mathematics Olympiad is open to all Primary 2, 3, 4, 5, 6, Secondary 1, 2 and 3 students in Singapore (or Grade 2, 3, 4, 5, 6, 7, 8 and 9 for International Schools). Primary 2 – 15 non-routine problems (1 point each), total 15 points. Primary 3 – 15 non-routine problems (1 point each), total 15 points. Primary 4 – 20 non-routine problems (1 point each), total 20 points. Primary 5 – 25 non-routine problems (1 point each), total 25 points. Primary 6 – 25 non-routine problems (1 point each), total 25 points. Secondary 1 – 25 non-routine problems (1 point each), total 25 points. Secondary 2 – 25 non-routine problems (1 point each), total 25 points. Secondary 3 – 25 non-routine problems (1 point each), total 25 points. • The duration of the AMO is 1 hour 30 minutes for each level. • Calculators are not permitted during the contest. • The aim of the AMO is to give more students an opportunity to perform mathematically on an international stage, and so to discover, encourage and challenge mathematically gifted students. Awards: Each participant will receive a Certificate of Participation or an award certificate and a medal if he/she wins a Gold/Silver/Bronze award. 1 Instructions for Using Answer Entry Sheet DIRECTIONS FOR MARKING ANSWER SHEET EXAMPLES OF SHADING 1. Use ONLY a black 2B lead pencil. DO NOT Use ink or ballpoint pen. 2. Shade circles of your choice completely. Erase cleanly any answer you wish to change. 3. Make no stray marks on this answer sheet. 4. DO NOT fold or staple this sheet. CORRECT WRONG 1. Name of Participant & School Name Write your name and school name in the given space. For example, if your name is James Smith and your school is Flower Primary School, write your name and school as follows: 2. How to shade the level Shade your level (grade) in the corresponding circle. For example, if your level is Primary 4, do as follows: 3. How to shade Index Number Write your 8-digit index number in the provided boxes and shade the corresponding circle. For example, if your Index Number is 12345678, do as follows: 2 4. How shade the answers for the questions Write your answer in the provided boxes and shade the corresponding circle. Case 1: The answer is a 1-digit number. Shade “0” for the tens and hundreds place. Example 1: 3 + 4 =? CORRECT WRONG Example 2: 3 × 3 =? CORRECT WRONG Case 2: The answer is a 2-digit number. Shade “0” for the hundreds place. Example 3: 30 − 7 = 23 Example 4: 7 × 12 = 84 3 CORRECT WRONG CORRECT WRONG Case 3: The answer is a 3-digit number. Example 5: 987 − 23 =? CORRECT WRONG Example 6: 9 × 12 = 108 CORRECT 4 WRONG American Mathematics Olympiad 2017 Primary 2 Contest Paper Name: ______________________________________________ School: ______________________________________________ INSTRUCTIONS: 1. Please DO NOT OPEN the contest booklet until the Proctor has given permission to start. 2. TIME : 1 hour 30 minutes 3. Attempt all 15 questions. Each question scores 1 point. No points are deducted for incorrect answers. 4. Write your answers neatly on the answer sheet. 5. PROCTORING : No one may help any student in any way during the contest. 6. No calculators are allowed. 7. All students must fill in your Name and School. 8. MINIMUM TIME: Students must stay in the exam hall at least 1h 15 min. 9. Students must show detailed working and put answers on the answer sheet. 10. No spare papers can be used in writing this contest. Enough space is provided for your working of each question. Remark: Counting numbers are whole numbers except 0, i.e. 1, 2, 3, 4, 5, … AMO 2017, Primary 2 Contest Question 1 What is the value of 10 + 9 + 8 + 7 + 6 − 5 − 4 − 3 − 2 − 1? Question 2 The sum of the digits of the number 108 is 9. What is the least number greater than 108 whose sum of the digits is 9? 6 AMO 2017, Primary 2 Contest Question 3 Find the value of 90 − 80 + 70 − 60 + 50 − 40 + 30 − 20 + 10. Question 4 Sandy multiplies 2 × 5 × 3 × 2 × 5 × 3 and writes the product as a whole number. What is the sum of the digits of this number? 7 AMO 2017, Primary 2 Contest Question 5 Summer vacation lasts for 30 days. During summer vacation, what is the greatest number of Fridays that could occur? Question 6 When I open my Maths book, two pages face me and the sum of the two page numbers is 217. What is the page number of the very next page? 8 AMO 2017, Primary 2 Contest Question 7 The first row has 2 dots: The second row has 4 dots: The third row has 6 dots: More rows are added. Each additional row has 2 more dots than the row before it. How many dots will there be in the first 8 rows combined? Question 8 When 24 is added to a number, the result is the same as when the number is multiplied by 3. What is the number? 9 AMO 2017, Primary 2 Contest Question 9 Dillon works at the park every third day. William works at the park every fourth day. They all worked together on Thursday. The park is open every day. On what day of the week will they all next work together? (If your answer is Monday, write 001; if it is Tuesday, write 002; and so on) Question 10 A certain book has its pages numbered from 1 to 30. Then any page number that contains the digit 2 is entirely erased. What is the sum of all of the numbers that were erased? 10 AMO 2017, Primary 2 Contest Question 11 Jack is John’s brother. The sum of their ages is 17. In 5 years Jack will be twice as old as John. How old will Jack be in 5 years? Question 12 The cost of 3 pencils and 2 markers is $12. The cost of 2 pencils and 3 markers is $13. What is the total cost of 6 pencils and 6 markers? 11 AMO 2017, Primary 2 Contest Question 13 At Springfield Elementary School, a shelf contains either 4 math books or 5 spelling books. On 20 shelves at Springfield Elementary, there are 65 spelling books and some number of math books. All 20 shelves are full. How many math books are there on the shelves? Question 14 Given: + + =8 + + =7 + + =5 What is the value of + + ? 12 AMO 2017, Primary 2 Contest Question 15 There are 84 unit cubes arranged in 4 square layers with no space between the cubes as shown. The layers are 1 by 1, 3 by 3, 5 by 5, and 7 by 7. How many unit cubes are completely surrounded by six other unit cubes? 13 AMO 2017, Primary 2 Contest Question 1 What is the value of 10 + 9 + 8 + 7 + 6 − 5 − 4 − 3 − 2 − 1? Solution Strategy: Use the idea that the numbers differ by a constant value. Regroup in the following way: 10 + 9 + 8 + 7 + 6 − 5 − 4 − 3 − 2 − 1 = (10 − 5) + (9 − 4) + (8 − 3) + (7 − 2) + (6 − 1) =5+5+5+5+5 = ππ Answer: 025 Question 2 The sum of the digits of the number 108 is 9. What is the least number greater than 108 whose sum of the digits is 9? Solution Strategy: Focus on the last two digits. Since the number has to be greater than 108, consider the hundreds digit to remain as “1.” The last two digits will then have to add up to 8. Since 1 + 7 = 8, the least number greater than 108 whose digit sum is 9 must be 117. Answer: 117 Question 3 Find the value of 90 − 80 + 70 − 60 + 50 − 40 + 30 − 20 + 10. Solution Strategy: Use convenient grouping. (90 − 80) + (70 − 60) + (50 − 40) + (30 − 20) + 10 = 10 + 10 + 10 + 10 + 10 = ππ Answer: 050 14 AMO 2017, Primary 2 Contest Question 4 Sandy multiplies 2 × 5 × 3 × 2 × 5 × 3 and writes the product as a whole number. What is the sum of the digits of this whole number? Solution Strategy: Use the associative and commutative properties. Insert parentheses to show grouping: (2 × 5) × 3 × (2 × 5) × 3 Reorder: 9 × 10 × 10 = 900. This has a digit sum of 9. Answer: 009 Question 5 Summer vacation lasts for 30 days. During summer vacation, what is the greatest number of Fridays that could occur? (If your answer is Monday, write 001; if it is Tuesday, write 002; and so on) Solution Strategy: Count the number of cycles of 7 days. We know that 30 days equals 4 weeks and 2 days. To maximize the number of Fridays, make the first day a Friday. The remaining 30 days will have 4 Fridays, so the greatest number of Fridays that could occur is 4 + 1 = π. Answer: 005 Question 6 When I open my Maths book, two pages face me and the sum of the two page numbers is 217. What is the number of the very next page? Solution Strategy: Find two consecutive numbers with sum of 217. 217 = 108 + 109. The very next page is 110. Answer: 110 15 AMO 2017, Primary 2 Contest Question 7 The first row has 2 dots: The second row has 4 dots: The third row has 6 dots: More rows are added. Each additional row has 2 more dots than the row before it. How many dots will there be in the first 8 rows combined? Solution METHOD 1: Strategy: Add the dots in the first 8 rows. Add the number of dots in all 8 rows: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 = ππ. [Note: Add using (2 + 16) + (4 + 14) + (6 + 12) + (8 + 10) = 18 × 4 = 72.] METHOD 2: Strategy: Visualize the dots in a familiar shape. Cut the shape into 2 equal parts and then rotate one of the parts about the other. The total number of dots is 8 × 9 = ππ. Answer: 072 Question 8 When 24 is added to a number, the result is the same as when the number is multiplied by 3. What is the number? Solution Strategy: Draw a Model Diagram: 1 unit 1 unit 24 1 unit 1 unit 2 units= 24 1 unit= 24 ÷ 2 = ππ Answer: 012 16 AMO 2017, Primary 2 Contest Question 9 Dillon works at the park every third day. William works at the park every fourth day. They all worked together on Thursday. The park is open every day. On what day of the week will they all next work together? (If your answer is Monday, write 001; if it is Tuesday, write 002; and so on) Solution Strategy: Calculate the first common multiple The multiples of 3 are 3, 6, 9, 12, 15, … , and the multiples of 4 are 4, 8, 12, 16, … Hence, the first common multiple (or least common multiple) of 3 and 4 is 12. Twelve days after Thursday is Tuesday, which is 002. Answer: 002 Question 10 A certain book has its pages numbered from 1 to 30. Then any page number that contains the digit 2 is entirely erased. What is the sum of all of the numbers that were erased? Solution Strategy: Use a pattern to find the sum. The erased numbers are 2, 12, 22 and also 20, 21, 22, 23, 24, 25, 26, 27, 28, 29. 2 + 12 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 = πππ Answer: 259 Question 11 Jake is John’s brother. The sum of their ages is 17. In 5 years Jake will be twice as old as John. How old will Jack be in 5 years? Solution Strategy: Draw model In 5 years, the sum of their ages will be 17 + 5 + 5 = 27. Jack: John: 27 So, each block (unit) is equal to 27 ÷ 3 = 9. In 5 years, Jack is 9 × 2 = 18 years old and John is 9 years old. Answer: 009 17 AMO 2017, Primary 2 Contest Question 12 The cost of 3 pencils and 2 markers is $12. The cost of 2 pencils and 3 markers is $13. What is the total cost of 6 pencils and 6 markers? Solution Strategy: Write a set of equations to find the cost. Let P = cost of a pencil and let M = cost of a marker. Then 3P + 2M = 12 and 2P + 3M = 13. Add these two equations together to get 5P + 5M = 25. Divide by 5 to get P + M = 5. Finally 6P + 6M = $30. Answer: 030 Question 13 At Springfield Elementary School, a shelf contains either 4 math books or 5 spelling books. On 20 shelves at Springfield Elementary, there are 65 spelling books and some number of math books. All 20 shelves are full. How many math books are there on the shelves? Solution METHOD 1: Strategy: Draw a diagram. Draw a diagram of twenty shelves. Place 5 spelling books on a shelf until 65 spelling books are placed. This requires filling 13 shelves because 13 × 5 = 65. Fill up the remaining shelves with 4 math books. 4 × 7 shelves = 28 math books. 5 spelling books 5 spelling books 5 spelling books 5 spelling books 4 math books 5 spelling books 5 spelling books 5 spelling books 4 math books 4 math books 5 spelling books 5 spelling books 5 spelling books 4 math books 4 math books 5 spelling books 5 spelling books 5 spelling books 4 math books 4 math books METHOD 2: Strategy: Find the number of shelves filled with spelling books. The 65 spelling books completely fill 65 ÷ 5 = 13 shelves. There are 20 − 13 = 7 remaining shelves to be filled. There are 4 × 7 = ππ math books that are used to fill them. Answer: 028 18 AMO 2017, Primary 2 Contest Question 14 Given: + + =8 + + =7 + + =5 What is the value of + + ? Solution METHOD 1: Strategy: Add and subtract equations to find the value of the circle. The desired amount has two circles and a square, so let’s double the first equation to find out that 2 triangles plus 2 squares plus 2 circles = 8 + 8 = 16. Now, the third equation has two triangles, so let’s subtract the third equation from this doubled one. (2 triangles + 2 squares + 2 circles) – (2 triangles + 1 square) is the same as 2 circles + 1 square, and that is the desired amount. This desired amount must be equal to 16 – 5 = 11. METHOD 2: Strategy: Find the value represented by each shape. We see from the second two equations that a square must be 2 more than a triangle. We also see by adding the last two equations together that 3 triangles + 3 squares = 7 + 5 = 12, so 1 triangle + 1 square equals 12 ÷ 3 = 4. Because a square is 2 more than a triangle and they add to 4, a triangle is 1 and a square is 3. This means that a circle is 8 − 1 − 3 = 4, and the desired amount is 2 × 4 + 3 = ππ. Answer: 011 19 AMO 2017, Primary 2 Contest Question 15 There are 84 unit cubes arranged in 4 square layers with no space between the cubes as shown. The layers are 1 by 1, 3 by 3, 5 by 5, and 7 by 7. How many unit cubes are completely surrounded by six other unit cubes? Solution Strategy: Make an organized table of information. The layers of the structure have been separated. None of the cubes in the top layer and bottom layer is completely surrounded by other cubes. The tops of the cubes that were completely surrounded by other cubes are shaded. Top cube exposed above 1 ο΄ 1 + 3 ο΄ 3 = 10 Bottom layer all exposed from below [Notice how the shading of one layer matches the number of cubes in the layer above it.] Answer: 010 20
