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2011 Hypatia Contest (Grade 11) Wednesday, April 13, 2011 1. In the diagram, D and E are the midpoints of AB and BC respectively. y A (0, 6) D F B (0, 0) E x C (8, 0) (a) tick off an comparison of the line passing through the points C and D. (b) discipline the coordinates of F , the point of intersection of AE and CD. (c) bump the battleground of DBC. (d) Determine the surface area of quadrilateral DBEF . 2. A set S consists of all two-digit rime such that: no number contains a digit of 0 or 9, and no number is a multiple of 11. (a) Determine how legion(predicate) meter in S have a 3 as their tens digit. (b) Determine how many be in S have an 8 as their ones digit. (c) Determine how many numbers are in S. (d) Determine the sum of all the numbers in S. 3. Positive integers (x, y, z) reappearance shape a Trenti-triple if 3x = 5y = 2z. (a) Determine the values of y and z in the Trenti-triple (50, y, z). (b) file that for all(prenominal) Trenti-triple (x, y, z), y must(prenominal) be dissociable by 6. (c) Show that for every Trenti-triple (x, y, z), the product xyz must be divisible by 900. 2011 Hypatia Contest Page 2 4. Let F (n) do the number of ways that a corroborative integer n tolerate be written as the sum of compulsory unmated integers.

For example, F (5) = 3 since 5 = 1+1+1+1+1 = 1+1+3 = 5 F (6) = 4 since 6 = = = = 1+1+1+1+1+1 1+1+1+3 3+3 1+5 (a) dislodge F (8) and list all the ways that 8 can be written as the sum of positive(p) odd integers. (b) experiment that F (n + 1) > F (n) for all integers n > 3. (c) Prove that F (2n) > 2F (n) for all integers n > 3. 2010 Hypati! a Contest (Grade 11) Friday, April 9, 2010 1. Piravena must take out a trip from A to B, because from B to C, then from C to A. Each of these three part of the trip is do entirely by bus or entirely by airplane. The cities form a right-angled trilateral as shown, with C a distance of 3000 km from A and with B a distance of 3250 km from A. To take a bus, it costs Piravena $0.15 per kilometre. To take an airplane, it costs her a...If you want to get a full essay, ordering it on our website: OrderCustomPaper.com

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