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Thursday, January 17, 2013

2013 Mathcounts State Prep: Harder State Questions

Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here.

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

Thursday, January 10, 2013

2013 Mathcounts Basic Concept Review: Some Common Sums/numbers

These are some common sums that appear on Mathcounts often.

1+2+3++n=n(n+1)2

 2+4+6+.2n=n(n+1)

1+3+5+.(2n1)=n2

The above are all arithmetic sequences.

The sum of any arithmetic sequence is average times terms (how many numbers).
Besides, the mean and median are the same in any arithmetic sequence.
Combining these knowledge, along with distributive rules some times
(case in point, sum of multiples of n, etc...) will expedite the calculation.

The "nth" triangular number is n(n+1)2

The sum of the first n triangular numbers is n(n+1)(n+2)6.

 12+22+32++n2 = n(n+1)(2n+1)6

13+23+33++n3=[n(n+1)2]2