Weekly Dose 17 - Maths Olympiad Practice

Weekly Dose 17 - Maths Olympiad Practice

• 1.
  Each duck weightsthe same, and each duckling weighs the same. If the total weight of 3 ducks and 2 ducklings is 32 kilograms, the total weight of 4 ducks and 3 ducklings is 44 kilograms, what is the total weight, in kilograms, of 2 ducks and 1 duckling? Solution: Let the weight of a duck as 𝑥 𝑘𝑔 and weight of a duckling as 𝑦 𝑘𝑔 3𝑥 + 2𝑦 = 32 −− − ① 4𝑥 + 3𝑦 = 44 −− − ② ① × 3 9𝑥 + 6𝑦 = 96 −− −③ ② × 2 8𝑥 + 6𝑦 = 88 −− −④ ③ − ④ 𝑥 = ____ 𝑘𝑔 From ①, 𝑦 = ____ 𝑘𝑔 ∴ 2𝑥 + 𝑦 = ____ 𝑘𝑔 Answer: 20 𝑘𝑔
• 2.
  A wooden rectangularblock, 4 cm x 5 cm x 6 cm, is painted red and then cut into several 1 cm x 1 cm x 1 cm cubes. What is the ratio of the number of cubes with two red faces to the number of cubes with three red faces? Solution: Only the cubes at the 8 corners of the block have three red faces. The cubes with two red faces lie along the 12 edges of the block (excluding the corners): - 4 of them have 4 cubes = 16 cubes; - 4 of them have 3 cubes = 12 cubes; - 4 of them have 2 cubes = 8 cubes ⇒ total of cubes with two faces = 36 The ratio is = 36 : 8 = __ : __ Answer: 9 : 2
• 3.
  What is thesmallest amount of numbers in the product 1 × 2 × 3 × 4 × ⋯ × 26 × 27 that should be removed so that the product of the remaining numbers is a perfect square? Solution: Rewrite the equation after prime factorization: 1 × 2 × 3 × 4 × ⋯ × 26 × 27 = 223 × 313 × 56 × 73 × 112 × 132 × 17 × 19 × 23 To get a square number of an integer, the power of the factors must be even. Therefore, we need to remove 2 × 3 × 7 × 17 × 19 × 23, 𝑂𝑅 remove 6 × 7 × 17 × 19 × 23, 𝑂𝑅 remove 3 × 14 × 17 × 19 × 23, 𝑂𝑅 remove 2 × 21 × 17 × 19 × 23, 𝑂𝑅 The smallest amount of numbers to be removed is ___ Answer: 5
• 4.
  We need tofind 𝑥(𝑥+1) 2 < 2008 < (𝑥+1)(𝑥+2) 2 From calculation, we know that 62×63 2 = 1953 < 2008 < 63×64 2 = 2016 Solution: In the figure below, the positive numbers are arranged in the grid follow by the arrows’ direction. For example, “8” is placed in Row 2, Column 3 and “9” is placed in Row 3, Column 2. Which Row and which Column that “2008” is placed? In Row 1, the 3rd number is 6 = 1+2+3, the 5th number is 15 = 1+2+3+4+5 Therefore, where the number of term is odd, the number in Row n is: 1+2+3+…+n = 𝑛(𝑛+1) 2 In Column 1, the 2nd number is 3 = 1+2, the 4th number is 10 = 1+2+3+4 Therefore, where the number of term is even, the number in Column n is: 1+2+3+…+n = 𝑛(𝑛+1) 2 Answer: Row 9, Column 55 2008 is located at Column (2008 – 1954) + 1 = 55, and Row (63 – 55) + 1 = 9