New Light Math

Q: If x is between 0 and 1, what is the probability that the greatest  integer ≤ log n (1/x) is odd, in terms of n. Definition of log n (1/x) = y is n y = 1/x. S: For the greatest integer ≤ log n (1/x) to equal 1, we must have a value  between 1/n and 1/n 2 . Then for the greatest integer ≤ log n (1/x) to equal 3, we must have a value between 1/n 3 and 1/n 4 . This process continues and we  are left with a geometric sequence:    (n-1)/n 2 + (n-1)/n 4 + (n-1)/n 6 + (n-1)/n 8 ….. = (n-1)/n 2    (n 2 -1)/n 2 = __ (n-1)__    (n+1)(n-1) =  _ 1 _   (n+1) Q: What is the remainder when (1)(1+2)(1+2+3)(1+2+3+4)...(1+2+3..+58+59) is divided by 61? S: We can rewrite this as:  1*2*2*3*3*4*4*5*5*6*...*58*59*59*60 2^59 We can then change this to: 59!*60!   2^59 By Wilson's formula, for any prime p , (p-1)! + 1 ≡ 0 (mod p) We can then ...

January 1 Intermediate Using an ordered alphabet of 26 letters, how many ways are there to chose a set of six different letters such that no two letters in the set are adjacent in the alphabet? For instance, { ISOKAY } is a valid set of six letters, but { V E TOI F } is not because E and F are both in the set. Solution: We can create a row of 27 boxes, the first empty and the rest filled with the letters A to Z in alphabetical order. We can then get 6 boxes of 2 with one filled and one empty, the filled one representing the letter we chose and the empty one to not chose any letter of the ones we have chosen before. We can see that we also need 15 other boxes to represent the non chosen letters not adjacent to the ones we chose. These boxes total to 6 + 15 = 21 and we must chose 6 or 15 of them. We get: 21*20*19*18*17*16  = 21*19*17*8 = 54264 ways. 6*5*4*3*2*1

Sum of cubes and Difference of Cubes a 3  + b 3  = (a + b) (a 2  − a b + b 2  ) a 3  - b 3  = (a - b) (a 2  + a b + b 2  ) Powers to know 5 7  = 78125 5 6  = 15625 5 5  = 3125 Compound Interest A = P(1+r/n) nt Harmonic Series 1+1/2+1/3+1/4+1/5+1/6...+1/n To estimate sum, take the 2 0  term for 1, then take the next 2 1  terms for 2, then take the next  2 2  terms for 3.... Sums: 2 - up to 1/4 2.5 - up to 1/7 3 - up to 1/11 3.5 - up to 1/19 4 - up to 1/31 4.5 - up to 1/51 5 - up to 1/83 Area of Octogon A = 2(1+√2) a 2 Finding the number sides of a regular polygon when you know the interior angle in degrees 180-x° = 360/n Finding total number of diagonals in a polygon n(n-3)/2

Missed on Sprint 11813: On Saturday, Wes rode his bike 15 miles in 70 minutes. On Sunday, Wes rode his bike 13 miles in 50 minutes. What was Wes’s average speed for his weekend of biking, in miles per hour? 14 mph 15+13=28 70+50=120 = 2 hours 28/2=14mph I got this wrong since I did 13+15=18 ← a very silly mistake that I should have never done On his first trip, Bob walked 12 miles in 100 minutes. On his second trip, he walked only 2 miles in 20 minutes. What is his average speed for the two trips? 12+2=14 100 + 20 = 120 = 2 hours 14/2 = 7 mph Going to work, Bob took a train that went 5 miles in 15 minutes. Coming back, he biked instead taking 30 minutes. What was his average speed for the two trips? 5+5=10 30+15=45=¾ hour 10/(¾)= 40/3 mph Cole scored 88, 91, and 94 on his first three exams, while Carly scored 93, 98, and 97 on her first three exams. They both took a fourth exam, and Carly’s average score on the four exams was 2 points higher than Cole’s average score on the four exams....

Q24 - David wants to arrange the twelve numbers from 1 to 12 in a circle so that any two neighboring numbers differ by either 2 or 3. Which of the following pairs of numbers have to be neighbors? A)  5 and 8            B)  3 and 5            C)  7 and 9          D)  6 and 8            E)  4 and 6 S24 - First, we know that 1 and 12 must be directly across from each other. Next, we find that the numbers next to 1 are 3 and 4 while the numbers next to 12 are 9 and 10. Then, we place the remaining numbers on there and see that, with trial and error, 3 and 9 must be on one side and 4 and 10 must be on the other. Then, we can place the remaining numbers to get a working circle: 1 3        4 6                  2 8                 ...