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Intermediate Math Circles
February 13, 2013
Contest Preparation 2 Selected Solutions
1. This was Question A2 on the 2011 Canadian Senior Math Contest
2. This was Question A4 on the 2011 Canadian Senior Math Contest
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3. For the solution to this problem, view the video from Intermediate Math Circles February
13, 2013.
4. Let the three consecutive multiples of 3 be a − 3, a and a + 3. We can quickly verify that
the average of these three numbers is a.
Since the average of the four consecutive multiples of 4 is (a + 27), let the numbers be
a + 21, a + 25, a + 29 and a + 33. Again, we can quickly verify that the average of these
four numbers is a + 27.
Clearly, the smallest number is a−3 and the largest number is a+33. We want the average
of these two numbers to be 2022, so:
a − 3 + a + 33
= 2022
2
2a + 30
= 2022
2
a + 15 = 2022
∴ a = 2007
It is straightforward to check this result. The three multiples of three are 2004, 2007 and
2010, and the average is 2007. The four multiples of 4 are 2028, 2032, 2036, and 2040.
The average is 2034 which is a + 27. Finally, the smallest number is 2004 and the largest
is 2040 with an average of 2022 as required.
5. This was Question 2 on the 2007 Fryer Contest
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5. Continued part (c)
6. This was Question 3 on the 2007 Galois Contest
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7. This was Question 24 on the 2007 Cayley Contest
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