Thursday, January 17, 2013

2013 Mathcounts State Prep: Harder State Questions

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2004 Mathcounts State Sprint #19: The points (x, y) represented in this table lie on a straight line. The point (13, q) lies on the same line. What is the value of p + q? Express your answer as a decimal to the nearest tenth. 
 
Solution: 
#19:  Look at the table and you'll see each time x + 2, y would -3. 
-5 to -14 is (-9), three times (-3) so p = 2 + 3 x 2 = 8
p would = 13 when 2 + 5.5 * 2 = 13 so q = -5  +  (5.5) * (-3) = -21.5
p + q = -13.5

2004 Mathcounts State Sprint #24: The terms x, x + 2, x + 4, ..., x + 2n form an arithmetic sequence, with x an integer. If each term of the sequence is cubed, the sum of the cubes is - 1197. What is the value of n if n > 3?
Solution: 
The common difference in that arithmetic sequence is 2 and the sum of the cubes is -1197, which means that these numbers are all odd numbers. (cubes of odd numbers are odd and the sum of odd terms of odd numbers is odd. )

(-5)3 + (-7)3 + (-9)3 = -1197  However, n is larger than 3 (given) so the sequence will look like this:
 (-9)3+ (-7)3+ (-5)3+ (-3)3+ (-1)3 + (1)3 + (3)3 = -1197
 x = - 9 and x + 2n = 3, plug in and you get -9 + 2n = 3;  n = 6

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