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Weekly Dose 13 - Maths Olympiad Practice

Weekly Dose 13 - Maths Olympiad Practice

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Weekly Dose 13 - Maths Olympiad Practice

  1. 1. Two bikers A and B were 370km apart traveling towards each other at a constant speed. They started at the same time, meeting after 4 hours. If biker B started 1 2 hour later than biker A, they would be 20km apart 4 hours after A started. At what speed was biker A traveling? Solution: B can cover 20km in 1 2 hour, therefore B’s speed = 20 ÷ 1 2 = 40km/h When they started at the same time and met after 4 hours, B has traveled 4 × 40 = 160𝑘𝑚. A has travelled 370 − 160 = 210𝑘𝑚. Therefore, A’s speed = 210 ÷ 4 = ____𝑘𝑚/h. Answer: 52 1 2 𝑘𝑚/ℎ
  2. 2. There are 5 circles with 3 different diameters. Some of the circles touch each other as shown in the figure below. If the total area of the unshaded parts is 20cm2, find the total area of the shaded parts, in cm2. Solution: Let’s name the radius of the smallest circle as 𝑎 and the area (let’s name it 𝑆) as 𝑎2 𝜋. The radius for big circle is 3𝑎 and the area (let’s name it 𝐵) is 9𝑎2 𝜋. The radius for medium circle is 2𝑎 and the area (let’s name it 𝑀) is 4 𝑎2 𝜋. The area of unshaded parts + the area of shaded parts = 𝐵 −−→ ① Given the area of unshaded parts = 𝐵 – 𝑀 + 2𝑆 – 𝑆 = 20𝑐𝑚2 𝐵 – 𝑀 + 𝑆 = 20𝑐𝑚2 9𝑎2 𝜋– 4𝑎2 𝜋 + 𝑎2 𝜋 = 20𝑐𝑚2 6𝑎2 𝜋 = 20𝑐𝑚2 −−→ ② From ①, the area of shaded parts = 𝐵 – area of unshaded parts = 9𝑎2 𝜋– 6𝑎2 𝜋 = 3𝑎2 𝜋 From ②, = ____𝑐𝑚2 Answer: 10 𝑐𝑚2
  3. 3. Which number should be removed from: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 so that the average of the remaining numbers is 6.1? Solution: The sum of eleven numbers is = ____ −−→ ① The average of remaining ten numbers is 6.1, their sum must be = 6.1 × 10 = 61 Hence the number which is removed is ① − 61 = ____ Answer: 5
  4. 4. Solution: The houses in a street are located in such a way that each house is directly opposite another house. The houses are numbered 1, 2, 3, … up one side, continuing down the other side of the street. If number 37 is opposite number 64, how many houses are there in the street altogether? The difference between the last number (n) and 64 = the difference between 37 and 1 𝑛 − 64 = 37 − 1 𝑛 = ____ Answer: 100

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