Weekly Dose 16 - Maths Olympiad Practice
- 1. Train A and Train B travel towards each other from Town a and Town b respectively, at a constant speed. The two towns are 1320km apart. After the two trains meet, Train A takes 5 hours to reach Town b while Train B takes 7.2 hours to reach Town a. How many km does Train A run per hour? Solution: 𝑥 ∶ 5 = 7.2 ∶ 𝑥 𝑥2= 7.2 × 5 = _____ 𝑥 = _____ Speed for A = 1320 ÷ 𝑥 + 5 = _____ km/h Answer: 120𝑘𝑚
- 2. In the figure below, in a right-angled triangle ACD, the area of shaded region is 10 cm2. AD = 5 cm, AB = BC, DE = EC. Find the length of AB, in cm. Solution: Because DE = EC, ∆𝐵𝐸𝐷 = ∆𝐵𝐸𝐶 = 10 𝑐𝑚2 Because AB = BC and DE = EC, AD = 2 × BE Since AD = 5cm, BE = 2.5 cm ∆𝐵𝐸𝐶 = 1 2 × BE× BC = 10 cm2 1 2 × 2.5 × BC = 10 cm2 BC = 8 cm AD = _____ cm Answer: 8 cm
- 3. Eve said to her mother, “If I reverse the two-digits of my age, I will get your age.” Her mother said, “Tomorrow is my birthday, and my age will then be twice your age.” It is known that their birthdays are not on the same day. How old is Eve? Solution: If Eve’s age is 𝑎𝑏, her mother’s age is 𝑏𝑎. 𝑎 and 𝑏 is whole number between 1 and 9 And 𝑏𝑎 + 1 = 2 × 𝑎𝑏. 10𝑏 + 𝑎 + 1 = 2 × (10𝑎 + 𝑏) 8𝑏 = 19𝑎 − 1 ---- ① Because all the multiples for 8 are even numbers, 19𝑎 must be odd number. If 𝑏 is the biggest number 9, 8𝑏 = 72, which mean 19𝑎 − 1 cannot be greater than 72, 19𝑎 cannot be greater than 73, 𝑎 cannot be greater than 73 19 which is 3 3 19 . When 𝑎 = 1, cannot fulfill ①, so 𝑎 cannot be 1 When 𝑎 = 3, 𝑏 = ____ Answer: Eve is 37 years old
- 4. Balls of the same size and weight are placed in a container. There are 8 different colors and 90 balls in each color. What is the minimum number of balls that must be drawn from the container in order to get balls of 4 different colors with at least 9 balls for each color? ** Note: Always treat this kind of question as finding worst case scenario, the bad luck case ** Let’s find the largest number of balls we can drawn without achieving the desired result. We may draw all 90 balls of each of 3 colors ⇒ 3 × 90 = 270 Then we may drawn 8 balls of each of the remaining colors ⇒ 5 × 8 = 40 If we draw one more ball, unavoidable the desired result will be met. Therefore by drawing 270 + 40 + 1 = ___ balls, we are guaranteed to get at least 9 balls of each of 4 colors. Answer: 311 Solution: