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Weekly Dose 18 - Maths Olympiad Practice

- 1. A giraffe lives in a right-angled triangle. The base and the height of the triangle are 12m and 16m respectively. The area is surrounded by a fence. The giraffe can eat the grass outside the fence at a maximum distance of 2m. What is the maximum area outside the fence, in which the grass can be eaten by the giraffe, in m2? (Given 𝜋 is 3.14 and answer to 2 decimal places.) Solution: Using the Pythagoras theorem, the hypotenuse of the enclosure is 122 + 162 = 20 The area which the giraffe can eat consists of three rectangles: 12 × 2, 16 × 2 and 20 × 2, as well as three circular sectors which together form a circle with radius 2m. Hence the area eaten = 12 × 2 + 16 × 2 + 20 × 2 + 𝜋 × 22 = ____ Answer: 108.56 𝑚2
- 2. If 25% of the people who were sitting stand up, and 25% of the people who were standing sit down, then 70% of the people are standing. How many percent of the people were standing initially? Solution: Let assume at first A people were sitting, and B people were standing. Let say C people are standing now. 𝐶 = 25% × 𝐴 + 75% × 𝐵 = 70% × (𝐴 + 𝐵) 25% × 𝐴 + 75% × 𝐵 = 70% × 𝐴 + 70% × 𝐵 5% × 𝐵 = 45% × 𝐴 𝐵 𝐴 = 45% 5% Or I can write it as 𝐵 ∶ 𝐴 = 9 ∶ 1 ∴ Percentage of people standing initially = 𝐵 𝐴+𝐵 × 100% = ___ Answer: 90%
- 3. A sedan of length 3 metres is chasing a truck of length 17 metres. The sedan is travelling at a constant speed of 110 kilometres per hour, while the truck is travelling at a constant speed of 100 kilometres per hour. From the moment when the front of the sedan is level with the back of the truck to the moment when the front of the truck is level with the back of the sedan, how many seconds would it take? Solution: The difference of speed is 110𝑘𝑚/ℎ – 100𝑘𝑚/ℎ = 10𝑘𝑚/ℎ = 10× 1000 60 ×60 = 100 36 𝑚/𝑠𝑒𝑐𝑜𝑛𝑑. The total distance to overtake the truck is 20m Time taken = 20 ÷ 100 36 = ___ 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 Answer: 7.2 seconds
- 4. Solution: There are three prime numbers. If the sum of their squares is 5070, what is the product of these three numbers? 𝑎2 + 𝑏2 + 𝑐2 = 5070 where 𝑎, 𝑏 and 𝑐 is prime number The unit digit of prime numbers are: 1, 2, 3, 5, 7, 9 The unit digit of the square of any prime numbers are 1, 4, 5, 9 To have the sum of the square of three prime numbers to have 0 as the unit digit, the unit digit of the square of three prime numbers must be 1, 4, 5. The prime number that have a square with unit digit 4 can only be 2 The prime number that have a square with unit digit 5 can only be 5 5070 = 22 + 52 + 𝑐2 𝑐2 = 5041 𝑐 = 71 The product of 2 × 5 × 71 = ____ Answer: 710

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