1. January 28, 2010 – Evening Session Why is today perfect?

2.  General Mathcounts Discussion… getting ready for Saturday, February 6 th !  Counting Warm-ups (10 minutes)  Sprint Practice (5 problems)  Target Practice (2 pairs)  Team Practice (5 problems)

3.  How many license plates consisting of 9 capital letters contain at least 6 Ms? Source: http://www.nctm.org/publications/mt.aspxhttp://www.nctm.org/publications/mt.aspx

5.  (a) How many 6-card poker hands can be formed that contain 3 cards from each of 2 different suits?  (b) How many 6-card poker hands can be formed that contain 4 cards from one suit and 2 cards from another?

7.  How many ways are there to break 10 students into 1 group of 4 and 2 groups of 3?

9.  5 problems (#11 – 15 in Greatest Problems)  6 2/3 minutes

10.  2 pairs (#60 - 63 in Greatest Problems)  12 minutes

11.  5 problems (#92- 96 in Greatest Problems)  10 minutes

12.  2651. A pattern is formed by the values obtained from the expression under the radicals. Note that the sequence formed is 5, 11, 19, 29, … The first of these is 5 = 1 + 4; the second is 11 = 1 + (4 + 6); the third is 19 = 1 + (4 + 6 + 8); and the fourth is 29 = 1 + (4 + 6 + 8 + 10). We need to find the value of the 50 th radical were this pattern to continue. That value is 1 + (4 + 6 + 8 + … + 102). The sum of the expression in the parentheses is 2(2 + 3 + 4 + … + 51). According to Gauss’s formula, the sum of the first 51 integers is 1326. So we solve 1 + (4 + 6 + 8 + … + 102) = 1 + 2(1326 – 1) = 2651

13.  Does the product of four consecutive integers always yield one less than a perfect square?  YES!!!!  Why? Algebraic Proof: If we let n represent the first of the four consecutive integers, then the product is n(n + 1)(n + 2)(n + 3) = n^4 + 6n^3 + 11n^2 + 6n. When 1 is added to this expression it can be factored to (n^2 + 3n + 1)^2, which is a perfect square.

14.  20 cents  An advanced approach involves creating a formula to determine the merchant’s income in terms of the price p. For each cent above 15 cents, the merchant’s sales decrease by 16; in terms of p, the merchant’s sales are 400 – 16(p – 15) = 640 – 16p. Since income is number of peaches sold times the price, the merchant’s income can be represented by the expression p(640 – 16p) = 640p – 16p 2 The vertex of the parabola is (20, 6400)…