May11-12Gr910Booklet-PS

CEMC at Home Grade 9/10 - Monday, May 11, 2020 Contest Day 2 Today's resource features two questions from the 2020 CEMC Mathematics Contests. 2020 Canadian Team Mathematics Contest, Team Problem #9 How many times does the digit 0 appear in the integer equal to 20 10 ? 2020 Canadian Team Mathematics Contest, Individual Problem #7 Twenty-seven unit cubes are each coloured completely black or completely red. The unit cubes are assembled into a larger cube. If 1 3 of the surface area of the larger cube is red, what is the smallest number of unit cubes that could have been coloured red? More Info: Check out the CEMC at Home webpage on Thursday, May 21 for solutions to the Contest Day 2 problems.

CEMC at Home Grade 9/10 - Monday, May 11, 2020 Contest Day 2 - Solution Solutions to the two contest problems are provided below, including a video for the second problem. 2020 Canadian Team Mathematics Contest, Team Problem #9 How many times does the digit 0 appear in the integer equal to 20 10 ? Solution: By factoring and using exponent rules, we have 20 10 = (2 × 10) 10 = 2 10 × 10 10 . Therefore, 20 10 = 1024 × 10 10 , which is the integer 1024 followed by ten zeros. Thus, 20 10 has eleven digits that are 0. That is, 10 zeros at the end and one coming from the 1024 at the beginning. 2020 Canadian Team Mathematics Contest, Individual Problem #7 Twenty-seven unit cubes are each coloured completely black or completely red. The unit cubes are assembled into a larger cube. If 1 3 of the surface area of the larger cube is red, what is the smallest number of unit cubes that could have been coloured red? Solution: Since 3 √ 27 = 3, the dimensions of the larger cube must be 3 × 3 × 3. Therefore, each side of the larger cube has area 3 × 3 = 9. A cube has 6 faces, so the total surface of the cube is made up of 9 × 6 = 54 of the 1 by 1 squares from the faces of the unit cubes. Since 1 3 of the surface area is red, this means 54 3 = 18 of these unit squares must be red. The unit cube at the centre of the larger cube has none of its faces showing, the 6 unit cubes in the centres of the outer faces have exactly 1 face showing, the 12 unit cubes on the edge but not at a corner have 2 faces showing, one of each of two adjacent sides, and the 8 unit cubes at the corners each have 3 faces showing. For any unit cube, there are either 0, 1, 2, or 3 of its faces showing on the surface of the larger cube. This means at most three faces of any unit cube are on the surface of the larger cube. Thus, there must be at least 6 cubes painted red in order to have 18 red unit squares on the surface of the larger cube. There are 8 unit cubes on the corners, so if we colour exactly 6 unit cubes red and the other 21 black, then arrange the cubes into a 3 × 3 × 3 cube so that the 6 red unit cubes are at the corners, there will be exactly 18 of the unit squares on the surface coloured red. Therefore, the answer is 6. Video Visit the following link for an explanation of the solution to the second contest problem: https://youtu.be/K9ax9uQESME

CEMC at Home Grade 9/10 - Tuesday, May 12, 2020 Shady Circles For each problem, use the information and diagram given to find the area of the shaded region . Express your answers as simplified exact numbers. For example, π + 1 and 1 - √ 2 are simplified exact numbers. Problem 1 Two circles are centred at C and O as shown. AB is a diameter of the larger circle. OB is a diameter of the smaller circle. The larger circle has a diameter of 20. Problem 2 The circle with centre O has a radius of 2. Points A and on the circle and ∠ AOB = 90 ◦ as shown. 2 Problem 3 Square ABCD is inscribed in the circle with centre O and radius 5 as shown. A square is inscribed in a circle if all four vertices of the square lie on the circle. 2 5 Problem 4 Two circles, each with a radius of 10, are centred at C and O as shown. The circles intersect at points A and B with ∠ AOB = 60 ◦ . Can you visualize what this diagram would look like if you pulled the circles apart? More Info: Check out the CEMC at Home webpage on Tuesday, May 19 for a solution to Shady Circles. To review area calculations involving circles and triangles, visit the following videos in the CEMC courseware: area of triangles and area of circles .