Vancouver Math Olympiad

Transcription

1 Vancouver Math Olympiad Competition #1: November 21 st, NWSS Library Game #1: Hands-on Geometry This game consists of both a theoretical as well as a practical portion. Your group will be given a pre-constructed regular octahedron with edge length 15cm. A regular octahedron has eight equilateral triangular faces. Your team will need to complete the following tasks: Theoretical portion: (1) Is there a path that goes on all 12 edges of the octahedron such that the path begins and ends on the same vertex? If so, indicate the path on the octahedron below, otherwise prove that such a path does not exist. (2) Compute the volume of the given (3) Compute the surface area of the given (4) Determine the measure of ACB in the following diagram of a regular (5) What s the distance from the center of one face of the octahedron to the center of the opposite face of the given octahedron? BONUS: (4 points) Determine the radius of the inscribed sphere of the given

2 Practical portion: (1) Draw a net for an (2) Using the Rubik s snake given (a staff member will walk around giving every team an opportunity to use the Rubik s snake), create an (3) Build a cylinder from the materials given such that it fits inside the octahedron of side length 15cm while attempting to maximize its volume. Calculate the volume of your constructed solid. (4) For this section of the practical portion, you will be given a regular octahedron with edge length 10cm, and the octahedron will henceforth refer to a regular octahedron with edge length 10cm. Build a cone from the materials given such that the octahedron fits inside it while attempting to minimize its surface area. Calculate the surface area of your constructed solid. Hint: for (3) and (4) of the practical portion, DO NOT attempt to calculate the theoretical maximum and minimum of the requested volume and surface area, respectively. Calculating it requires mathematics beyond high school level, and will only waste your team s time. Try to eyeball or make an educated guess on where the cylinder or cone should be placed relative to the octahedron to achieve the maximum volume or minimum surface area respectively. Evaluation: For the theoretical portion, each problem is worth 4 points. Full marks will be awarded for a correct answer. However, any work shown will be considered for partial marks. Answers need not be in exact form, however, rounded answers must have exactly 4 decimal places. For the practical portion, (1) and (2) are worth 2 points each, and for (3) and (4), each constructed shape is assessed on three levels: Cylinderness/Coneness: (4 points) How closely does the solid resemble an actual cylinder or cone? Both of your solids should be able to roll on their sides if the judges apply a gentle push to each. Volume/Surface Area Optimization: How close is the volume/surface area of the solids to the theoretical optimized answer? (8 points) Does the cylinder fit inside the larger octahedron? Does the smaller octahedron fit inside the cone? (minus 2 points for each violation) Calculations: (4 points) Are the calculations of the volume/surface areas of your solids correct? Total: 20 points for theoretical portion + 20 points for practical portion = 40 points

3 Theoretical portion: (4 points each 5 = 20 points) (1) Is there a path that goes on all 12 edges of the octahedron such that the path begins and ends on the same vertex? If so, indicate the path on the octahedron below, otherwise prove that such a path does not exist. (2) Compute the volume of the given

4 (3) Compute the surface area of the given (4) Determine the measure of ACB in the following diagram of a regular

5 (5) What s the distance from the center of one face of the octahedron to the center of the opposite face of the given octahedron? BONUS: (4 points) Determine the radius of the inscribed sphere of the given