CAPCON zoom session -- Number Sense

The following questions were sent to me for the UIL problem solving zoom session on July 10. There was not enough time to get to all of the questions I received. I put together the following ideas, hints, and suggestions for the problems I was unable to get to. I will be submitting a similar posting for math problems. If you still need assistance with any of these feel free to email me. Mr. White

11 1 1 1 (??) 3 6 10  15  ... 55 = _____. ***** I did this one at the end of the zoom sessions and made an error . 55 is the 11th triangular number, not 10 as I stated. So, the sum of these triangular number reciprocals would be (11q 1 ) 10 5 (11 1) = 12 = 6 *****

(??) Find the sum of the first four triangular numbers. ______***** The first four triangular numbers are 1, 3, 6, & 10. The sum is easy given a small (n)(n+1)(n+2) set of numbers. There is a shortcut for larger sets: Sn = 6 *****

2020 HS B

(45) (201)3 = ______***** This is a great problem based on a binomial and/or Pascal triangle: (a b)33 = a 3a 2 b 3ab 2 b 3 If you let a = 200 and b = 1 ==> 20032 3(200 (1) 3(200)(1) 23  1 ==> 8000000  120000 600 + 1 . Now if you look at the place value you get 8 12 06 01 ex. (401)3 ==> 64 48 12 01 Problem could be extended to (x0y)3 *****

(49) (51)33q (52) = ______***** This particular problem is not an easy one to do in your head. Looking at it algebraically: (a)33q (a 1) will simplify to q (3a2  3a  1) ==> qq 3(a)(a 1) 1 ==> 3(51)(52)  1. A simpler and easier example would be: (30)33qq (31) ==> 3(30)(31) 1. Naturally, if it is (a q 1) 33 (a) it would be a positive result instead of a negative one. *****

2020 District

(33) Find the smallest k, where k 6, such that 5k 4 is a perfect cube. ______***** No magic here ... look at ending in 4 ***** (58) The length of the altitude to the of a 9' qq 40' 41' triangle is ______ft ***** Using a222 b = c you can solve for h. You should find a simplified formula involving a, b, & c to work these types of problems easily.

x

a c - x h

b

*****

2019 HS SAC

*(50) 8‚‚‚ 16 24 32 = ______***** Interesting to note these are multiples of 8 (could be multiples of anything). Since it is an estimation, there are various strategies. I like the following: 83 (1‚‚ 2 3) ‚ 32 ==> 512(6) ‚ 32 ==> about 3100 x 32 Could have done 834 (1‚‚ 2 4) ‚ 24 ==> 8 ‚ 24 Strategies have to be developed for estimation problems *****

2018 HS A

#48. The sum of the reciprocals of all the positive integral divisors of 20 is ____. 1111 1 1 105 4 2 211 ***** 12451020  = 1 202020202020 = 1 = 2 Take note of the denominators === and now the the numerators Try doing this with 6 or 12 or whatever. The shortcut is pretty clear *****

2018 HS B

#49. The sum of the reciprocals of all of the positive integral devisers of R is 1.444... . R = ___. 4 ***** From the previous problem note 1.444... = 19 ==> the divisors of 9 are 1, 3, 9 111 31 4 139 = 1  99 = 1 9 *****

#56. (259)(39)(k) = 121,212. k = ___. ***** expanding this to 3 x 7 x 13 x 37 = 10101 ==> 10101 x 12 = 121212. hmmm ... interesting set of primes *****

2018 HS Regional

#55. The probability of randomly selecting a composite number from the set of positive digits is ____%. ***** Probability = ratio of the correct outcomes to the total total outcomes Composite digits ==> 0, 4, 6, 8, 9 and digits ==> 0,1,2,3,4,5,6,7,8,9 ==> 5/10 ***** 2012 HS B

#13. 6/7 - 3/14 - 1/28 ***** no magic, just common denominator or 28 ==> (24 - 6 - 1)/28 *****

#51. 115 x 252 ***** no magic but a confident strategy needs to be developed. I break the problem up as follows: 115 ==> 1 11 115 15 5 x 252 ==> x 2 x 25 x 252 x 52 x 2 ------starting at the right ==> 2x5 = 10 ==> put down 0 carry 1 cross mult. ==> 2x1 + 5x5 + 1 ==> 28 ==> put down 8 carry 2 tri mult. ==> 2x1 + 5x1 + 2x5 + 2 ==> 19 ==> put down 9 carry 1 cross mult. ==> 5x1 + 2x1 + 1 ==> 8 ==> put down 8 on the left ==> 2x1 ==> 2 ==> put down the 2 = 28980 *****

#52. 9^10 / 11 has a remainder of ____. ***** One method is to expand to make finding the remainder easier: 922222 9 9 9 9 11 x 11 x 11 x 11 x 11 ==> remainders of 4 x 4 x 4 x 4 x 4 4 4 4 4 4 6416 ==> 11 x 11 x 11 x 11 x 11 ==> 11 x 11 ==> remainders of 9 x 5 45 ==> 11 ==> remainder 1 big hint: There is a really nice math theorem about remainders that makes these problems a snap *****

2012 HS District 1

#1. 5/8 - 4/7 = Is this just a straight calculation? ***** straight forward calculation *****

#47. Which of the following is NOT a triangular number, 105, 114 or 110? ***** A property of triangular numbers: T is triangular if 8(T) 1 results in a perfect . ex. 10 is triangular since 8(10) + 1 = 81 which is a square number *****

#54. If A is 40% more than B and C is 60% less than B, then C is what part of A? 725 ***** A = 552 B and C = B ==> B = C. Substitute in solve for C. *****

#58. (1 + 4 + 9 + 16 +..+49 + 64)/(1 + 3 + 6 + 10 +..+ 28 + 36)= ___ ***** notice that the numerator is the sum of the first 8 natural numbers squared and the denominator is the sum of the first 8 triangular numbers. The first problem at the top of the page I shared (n)(n+1)(n+2) Sn = 6 , the formular for finding the sum of the triangular numbers. There is a similar formula for the sum of the of the natural numbers. Find that formula and this problem is a snap. ***** 2012 HS District 2

#42. The sum of the first 4 triangular numbers is ___. ***** see other example already discussed *****

#43. 321 x 235 = ___. ***** see other example already discussed *****

2012 Regional

#41. 225 x 134 = (Wonder if we use 4 x 225 = 900?) ***** see other example already discussed *****

*#50 ((( of 5) + 1)/2) x 31.4 x 27.18. Estimation. ***** ... 1.618 x 31.4 x 27.18 ... develop a strategy *****

#52. 1/3 + 1/6 + 1/10 + 1/15 + ... + 1/36 ***** see other example already discussed ***** 2012 State

#50. 31.4 x + 27.1 x e + 16.1 x theta ***** 31.4 x 3.14 + 27.1 x 2.71 + 16.1 x 1.618 ... develop a strategy *****

#52. 1 + 3 + 6 + 10 + 15 + ...+ 78 ***** see other example already discussed *****

2011 HS Regional

#50. (((square root of 5)+1)/2)^2 (pi)^2 (e)^2 = ___. ***** see other example already discussed *****

#52. 141 x 232 = __ ***** see other example already discussed ***** 2011 State

#46. 323 x 414 = _. ***** see other example already discussed *****