Soal Olimpiade Matematika Junior Australia Tahun 2009

Western Australian

Junior Mathematics Olympiad 2009

Individual Questions                                                                       100 minutes

 

General instructions: Each solution in this part is a positive integer less than 100. No working is needed for Questions 1 to 9. Calculators are not permitted. Write your answers on the answer sheet provided.

 

1.    Evaluate

[1 mark]

 

2.    From each vertex of a cube, we remove a small cube whose side length is one-quarter of the side length of the original cube. How many edges does the resulting solid have?

[1 mark]

 

3.    A certain 2-digit number x has the property that if we put a 2 before it and a 9 afterwards we get a 4-digit number equal to 59 times x.

What is x?                                                                                                                        [2 marks]

 

4.    What is the units digit of 2²⁰⁰⁹ ´ 3²⁰⁰⁹ ´ 6²⁰⁰⁹?                                                           [2 marks]

 

5.    At a pharmacy, you can get disinfectant at di_erent concentrations of alcohol. For instance, a concentration of 60% alcohol means it has 60% pure alcohol and 40% pure water. The pharmacist makes a mix with 3/5 litres of alcohol at 90% and 1/5 litres of alcohol at 50%.

How many percent is the concentration of that mix?                                              [2 marks]

 

6.    If we arrange the 5 letters A, B, C, D and E in diferent ways we can make 120 diferent “words". Suppose we list these words in alphabetical order and number them from 1 to 120. So ABCDE gets number 1 and EDCBA gets number 120.

What is the number for DECAB?                                                                                [3 marks]

 

7.    Every station on the Metropolis railway sells tickets to every other station. Each station has one set of tickets for each other station. When it added some (more than one) new stations, 46 additional sets of tickets had to be printed.

How many stations were there initially?                                                                    [3 marks]

 

8.    At a shop, Alice bought a hat for $32 and a certain number of hair clips at $4 each. The average price of Alice's purchases (in dollars) is an integer.

What is the maximum number of hair clips that Alice could have bought?        [3 marks]

 

9.    The interior angles of a convex polygon form an arithmetic sequence:

143⁰, 145⁰, 147⁰, ….

How many sides does the polygon have?                                                                [4 marks]

 

10. For full marks, explain how you found your solution.

A square ABCD has area 64 cm2. Let M be the midpoint of BC, let d be the perpendicular bisector of AM, and let d meet CD at F. How many cm2 is the area of the triangle AMF?

 [4 marks]