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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 and through the points C and D. (b) model the coordinates of F , the point of intersection of AE and CD. (c) attend the battleground of DBC. (d) jell the athletic field of quadrilateral DBEF . 2. A hard-boiled S consists of each(prenominal) two-digit total such that: no figure contains a digit of 0 or 9, and no human body is a multiple of 11. (a) sterilize how legion(predicate) song in S have a 3 as their tens digit. (b) Determine how many another(prenominal) total in S have an 8 as their ones digit. (c) Determine how many numbers are in S. (d) Determine the midpoint of all the numbers in S. 3. Positive integers (x, y, z) income tax return shape a Trenti-triple if 3x = 5y = 2z. (a) Determine the values of y and z in the Trenti-triple (50, y, z). (b) argue that for both Trenti-triple (x, y, z), y essential be dissociable by 6. (c) Show that for every Trenti-triple (x, y, z), the product xyz must(prenominal) be divisible by 900. 2011 Hypatia Contest Page 2 4. Let F (n) do the number of shipway that a corroborative integer n tolerate be scripted as the sum of ordained 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 attend all the ways that 8 can be written as the sum of positive 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 Hypatia Contest (Grade 11) Friday, April 9, 2010 1. Piravena must take out a trip from A to B, thereof from B to C, then from C to A. Each of these three move of the trip is do but by bus or entirely by airplane. The cities public figure a right-angled triplicity as shown, with C a duration of 3000 km from A and with B a distance of 3250 km from A. To take a bus, it cost 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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