20142012加拿大滑铁卢大学11年级数学竞赛试题VIP免费

TIPS:1.Please read the instructions on the front cover of this booklet.2.Write all answers in the answer booklet provided.3.For questions marked, place your answer in the appropriate box in theanswer booklet and show your work.4.For questions marked, provide a well-organized solution in the answerbooklet. Use mathematical statements and words to explain all of the stepsof your solution. Work out some details in rough on a separate piece of paperbefore writing your finished solution.5.Diagrams are not drawn to scale. They are intended as aids only.1.For real numbers a and b with a ≥ 0 and b ≥ 0, the operation  is defined bya  b =√a + 4b.For example, 5  1 =5 + 4(1) =√9 = 3.(a) What is the value of 8  7?(b) If 16  n = 10, what is the value of n?(c) Determine the value of (9  18)  10.(d) With justification, determine all possible values of k such that k  k = k.2.Each week, the MathTunes Music Store releases a list of the Top 200 songs.A newsong “Recursive Case” is released in time to make it onto the Week 1 list.The song’sposition, P, on the list in a certain week, w, is given by the equation P = 3w 2 −36w +110.The week number w is always a positive integer.(a) What position does the song have on week 1?(b) Artists want their song to reach the best position possible. The closer that theposition of a song is to position #1, the better the position.(i) What is the best position that the song “Recursive Case” reaches?(ii) On what week does this song reach its best position?(c) What is the last week that “Recursive Case” appears on the Top 200 list?3.A pyramid ABCDE has a square base ABCD of side length 20. Vertex E lies on theline perpendicula...