2011 Fryer Contest (Grade 9)
3:41 AM
Muhammad Yusuf
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1. An arithmetic sequence is a sequence in which each term
after the first is obtained from the previous term by adding a constant d, called the common difference. For example, 2, 5, 8, 11, 14 are the first five terms of an arithmetic
sequence with a common difference of d = 3.
(a) Determine the 6th and 7th terms of the sequence given
above.
(b) What is the 31st term in this sequence?
(c) If the last term in this
sequence were 110, how many terms would there be in the sequence?
(d) If this sequence is
continued, does 1321 appear in the sequence? Explain why or why not.
2. In any isosceles triangle ABC with AB = AC, the altitude AD bisects the base BC so that
BD = DC.
(a) (i) As shown in ∆ABC, AB = AC = 25 and BC = 14. Determine the length of the altitude AD.
(ii) Determine the area of ∆ABC.
(b) Triangle ABC from part (a) is cut along its
altitude from A to D (Figure
1).
Each of the two new triangles is
then rotated 900 about point D until B meets C directly below D (Figure 2).
This process creates the new triangle which is
labelled PQR (Figure 3).
(i) In ∆PQR,
determine the length of the base PR.
(ii) Determine the area of ∆PQR.
(c) There are two different
isosceles triangles whose side lengths are integers and whose areas are 120.
One of these two triangles, ∆XYZ, is shown. Determine the lengths of the three
sides of the second triangle.
3. Begin with any two-digit
positive integer and multiply the two digits together. If the resulting product
is a two-digit number, then repeat the process. When this process is repeated,
all two-digit numbers will eventually become a single digit number. Once a
product results in a single digit, the process stops.
For example,
Note :
For first row : The process stops
at 8 after 3 steps.
For second row : The process
stops at 6 after 2 steps.
For third row : The process stops
at 0 after 1 step.
(a) Beginning with the number 68,
determine the number of steps required for the process to stop.
(b) Determine all two-digit
numbers for which the process stops at 8 after 2 steps.
(c) Determine all two-digit
numbers for which the process stops at 4.
(d) Determine a two-digit number
for which the process stops after 4 steps.
4. Ian buys a cup of tea every
day at Jim Bortons for $1.72 with money from his coin
jar. He starts the year with 365 two-dollar (200¢)
coins and no other coins in the jar. Ian makes payment and the cashier provides
change according to the following rules:
• Payment is only with money from the coin jar.
• The amount Ian offers the cashier is at least $1.72.
• The amount Ian offers the cashier is as close as
possible to the price of the cup of tea.
• Change is given with the fewest number of coins.
• Change is placed into the coin jar.
• Possible coins that may be used have values of 1¢, 5¢, 10¢, 25¢, and 200¢.
(a) How much money will Ian have
in the coin jar after 365 days?
(b) What is the maximum number of
25¢ coins that Ian could have in the
coin jar at any one time?
(c) How many of each type of coin
does Ian have in his coin jar after 277 days?
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