August, 2015 Calculators are PROHIBITED - The New York Times

[Pages:3]2015 Stanford-Math LeagMuAeTHTLoEuArGnUaEmPReEnSSt

Math League Press, P.O. BPo.Ox .1B7o,xT1e7n,aTfelny,afNly,eNweJwerJseersyey0707667700--00001177

August, 2015--Calculators are PROHIBITED "Speed Test 60" Answers & BRIEF SOLUTIONS--PAGE 1 of 3

Write your answer in the answer box to the right of each question. (4 points each)

Name:

Number Correct:

1. If my age 24 years ago was 24, I am now 48 and in 24 years I will be 72.

2. (2-1 + 3-1) -1 = (1/2 + 1/3) -1 = (5/6)-1 = 6/5.

3. 1 1 1 1 > 2, so n = 4. 1234

4. The product of 2015 different integers is even provided one of them is even; the 2014 other integers may all be odd.

5. 50 millimeters = 50/1000 meters = 0.05 meters 6. The average of 100 numbers, one of which is 99 and 99 of which are 100s, is

(99 + 99?100)/100 = 9999/100 = 99.99. 7. 1% of 3% of 10 = 0.01?0.03?10 = 0.003.

1. 72 2. 6/5 3. 4

4. 2014

5. 0.05 6. 99.99

7. 0.003

8. At the x-intercept of y = 2x + 4, y = 0. Therefore, 0 = 2x + 4 and x = -2.

9. If 2 = 2 , 4x + 6 = 9. Thus, 4x ? 1 = 2 and 1 1 .

4x 6 9

4x 1 2

10. If 2% of y is 3, then 0.02?y = 3. Solving, y = 3/0.02 = 150.

11. 1 x 3 . By inspection, x = 3. 1 1 x1 x1 x

12. 1 1 2 2 3

3

2

3

2

3

2

3 4 3 2 + 1

3.

13. Two different circles can intersect in at most 2 points.

14. On a digital clock, the time 1:23 forms an arithmetic sequence. In 12

minutes, the time will be 1:35 (another arithmetic sequence).

15. With 24 people in a room, 2 could be born in each of the 12 months. With 25

8. -2 9. 1

2 10. 150 11. 3

12. 5

13. False 14. 12

15. 25

people, there must be at least 1 month in which 3 or more were born. 16. The sum of any irrational number and its inverse is 0. 17. A diagonal divides the quadrilateral into 2 congruent triangles. Since

corresponding angles are equal, the opposite sides are parallel. 18. The slope of 2x + 3y = 1 is -2/3. Any line perpendicular will have slope 3/2.

19. If the sum of three prime numbers is 20, the sum could be 13 + 5 + 2. 20. Let w = the weight of my brain. Then 0.90 + 0.5w = w. Solving, w = 1.80.

16. False 17. True

18. 3/2 19. 13 20. 1.80

2015 Stanford-Math LeagMuAeTHTLoEuArGnUaEmPReEnSSt

Math League Press, P.O. BPo.Ox .1B7o,xT1e7n,aTfelny,afNly,eNweJwerJseersyey0707667700--00001177

August, 2015--Calculators are PROHIBITED "Speed Test 60" Answers & BRIEF SOLUTIONS--PAGE 2 of 3

Write your answer in the answer box to the right of each question. (4 points each)

Name:

Number Correct:

21. If x2 = x, x2 ? x = 0. Factoring, x(x ? 1) = 0. Thus, x = 0 or 1. If x = 0, 2-x = 2-0 = 1. 22. The probability that neither was born on a Monday is (6/7)?(6/7) = 36/49.

The probability that one was born on a Monday is 1 ? (36/49) = 13/49.

23. The longest line segment is the diagonal whose length is 12 12 12 3.

24. 2 radians = 2?180/ = 180 degrees.

25. Begin with $100. After 1 year it becomes $110. After 2 years, it is $121. That's a 21% increase from the beginning of year 1 to the end of year 2.

26. The diagonal of a 6?8 rectangle is 10, so the diagonals of the square are each 20. The area of the square is one-half the product of its diagonals.

27. There are 6 such numbers, so each digit appears 2 times in each place. The average of the digits in each place is 2, thus the overall average is 222.

28. Parallelogram P has a 30? angle and sides of length 5 and 10. P's area is 0.5?5?10 = 25. The area of an inscribed triangle is at most half of this.

29. (1010)2 1020 (102)10 10010.

30. There are 6 pairs of lines, so there are at most 6 distinct intersection points.

31. (xy)(yz)(xz) = (xyz) 2 = 9?6?24 = 1296, so xyz = ?36.

32. Since 31 < 1000 < 32, the largest such square is 312 = 961.

33. For integral n > 2, nn+1 > (n + 1)n. (Try some small values of n.) 34. From June 1 to October 1 is 122 days. When 122 is divided by 7, the

remainder is 3, so it's 3 days after Saturday. 35. Any isosceles triangle with only one side of length 2, and whose altitude to

That side has length 1 is a 45?-45?-90? triangle. The length of each leg is 2 .

36. x2 = |x|.

37. The least common multiple of 15 and 20 and 60. The greatest common factor of 15 and 20 is 5. Their difference is 55.

38. A pump fills an empty pool in 3 days, so it fills 2/3 of the pool in 2 days if working alone, and the 2nd pump, working alone, fills 1/3 of the pool in 2 days. Thus, the 2nd pump takes 6 days to fill the pool when working alone.

39. My teacher raised my grade of 50 by 20% to a 60. The principal then reduced my grade of 60 by 20% to a 48.

40. Since 6! = (3?2)?5?4?3?2 = (2?5)(3?3)(4?2) = 10?9?8, (6!)(7!) = 10!.

21. 1 22. 13

49 23. 3 24. False 25. 21

26. 200

27. 222

28. 12.5

29. True 30. 6 31. -36 32. 961 33. True 34. Tuesday

35. 2 + 2 2 36. False 37. 55

38. 6

39. 48

40. 10

2015 Stanford-Math LeagMuAeTHTLoEuArGnUaEmPReEnSSt

Math League Press, P.O. BPo.Ox .1B7o,xT1e7n,aTfelny,afNly,eNweJwerJseersyey0707667700--00001177

August, 2015--Calculators are PROHIBITED "Speed Test 60" Answers & BRIEF SOLUTIONS--PAGE 3 of 3

Write your answer in the answer box to the right of each question. (4 points each)

Name:

Number Correct:

41. The number of different numbers between 100 and 600 that can be formed using three different digits from the digits 1, 2, 3, 4, and 5 is 5?4?3 = 60.

42.: 2 3 3 2 (2 3 ) 2 (3 2 ) 2 2 6 32. But 2 6 < 23 < 9.

43. If (1 + r)2 + (1 ? r)2 = 20, 2r2 + 2 = 20. Thus, r = ?3 and (1 + 2r)2 = 49 or 25.

44. Draw a to the chord and a radius to an end of the chord forming a 3-4-5 .

45. A parallelogram inscribed in a circle has opposite angles both = and supplementary. 46. If a2 ? a = b2 ? b, then (a + b)(a ? b) ? (a ? b) = 0. Factoring, (a + b ? 1)(a ? b) = 0.

47. This is true in any arithmetic sequence whose 1st term is prime and whose

common difference is a multiple of that same prime, like 2, 4, 6, 8, 10, . . . .

48. This is a square, so its area is half the product of its diagonals.

49. The sum of 3 consecutive integers is always a multiple of 3. Of the first 100

positive integers, 33 are multiples of 3.

50. If the area of an equilateral triangle is 3 , the length of each its sides is 2.

51. If the angle-measures of a triangle form an arithmetic sequence, one angle must have measure 60. The other angles could have measures of 1 and 119.

52. 1 > 2015/2016 > 1/2016 + 1/2017 + 1/2018 + . . . + 1/4030 > 2015/4030 = 1/2.

A) 0 < S < 1/2

B) 1/2 < S < 1

C) S > 1

53. If x2 + y2 ? 2x + 4y ? 4 = 0, (x ? 1) 2 + (y + 2)2 = 32. The length of a radius is 3.

54. He recently lost 20 kg. If his new weight is w, then w = 0.9(w + 20). Hence,

0.1w = 18 and w = 180.

55. The difference is even if the two integers are both odd or both even. After

choosing the first integer, 4 of the remaining ones have the same parity. 56. The following products are perfect squares: 1?4, 1?9, 2?8, 4?9. There are

45 ways to choose two different integers, so the probability is 4/45.

57. Since r2 = 1/, r = 1/ and C = 2r = 2.

58. y = x is to x+y = 8 and passes through (3,3) and (4,4). The distance is 12 12 .

59. Since 8+12 = 20, it's a right triangle. Area = (1/2) ( 8)( 12) = (1/2) 96 24.

60. Group the fractions by denominators. Each such grouping has a sum of 1. After 13 groupings (91 terms), the sum is 13. Now add 1 nine times. 14

41. 60

42. False 43. 25 44. 3 45. True 46. False 47. True

48. 1/2 49. 33

50. 6 51. 119 52. B

53. 3 54. 180

55. 4 9

56. 4 45

57. 2 58. 2 59. 24 60. 13 9

14

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