1. Alabama School of Fine Arts
2019 Invitational Mathematics Tournament
Algebra 1 Ciphering

2. Points and collecting papers
Two minutes per question
Correct in the 1st minute = five (5) points
Correct in the 2nd minute = two (2) points
Incorrect in either minute = zero (0) points
A five-second warning will be given before the end of each interval.
For your paper to be collected, it must be folded and held high above your head.
Do not put your paper up until “Warning” is called.
If your paper is not in the air before the call of “Interval,” your paper will not be collected.
Runners will collect papers from behind you, not in front. Do not try to hand your paper to the runner in front of you.

3. Writing and formatting answers
All answers must be written in the answer box on your answer sheet.
Correct answers not in the box will not be counted.
No units except for money (dollar and cent signs only).
If an incorrect unit is included in the answer, it will be counted wrong.
All fractions must be in lowest terms.
Percent solutions must be written as a percent, not as the fraction or decimal equivalent.
All denominators must be rationalized.
No fractions under a radical
All radicals must be simplified.
All exponents must be positive.

4. Find the product of the solutions to the following equation:
Practice Question
Find the product of the solutions to the following equation:
𝑥 2 −5𝑥−6=0
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2nd Interval
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5 Seconds
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6. Practice Answer -6

7. Round 1

8. Question 1-1

9. Question 1-1
A standard deck of cards has four suits (♠♥♣♦). Each suit consists of 13 cards: number cards 2-10; three face cards J, Q, K; and an ace. In a standard deck of cards, what is the probability of drawing a King or a red card?
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2nd Interval
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11. Answer 1-1
7 13

12. Question 1-2

13. Question 1-2
How many ways can the letters in the word “GOOFIN” be arranged if the two O’s cannot be next to each other?
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2nd Interval
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15. Answer 1-2

16. Question 1-3

17. Question 1-3 If 𝑎➤𝑏=( 𝑎 2 − 𝑏 2 ), find 299➤298. 0:41 0:42 0:40 0:44
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19. Answer 1-3
597

20. Question 1-4

21. Question 1-4 Solve for x: (243 ) 1/3 = 3 4𝑥+1 0:41 0:42 0:40 0:44 0:45
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23. Answer 1-4
𝟏 𝟔

24. Question 1-5

25. Find the sum of the units digits of 1 9 20 and 1 8 19 .
Question 1-5
Find the sum of the units digits of and
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27. Answer 1-5 3

28. Round 2

29. Question 2-1

30. Simplify the following expression without any negative exponents:
Question 2-1
Simplify the following expression without any negative exponents:
[( 𝑤 6 𝑥 −8 𝑦 2 𝑧 −5 𝑤 4 𝑥 −2 𝑦 5 𝑧 −6 )-1]2
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2nd Interval
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32. Answer 2-1

33. Question 2-2

34. What is the sum of the odd positive integer factors 2022?
Question 2-2
What is the sum of the odd positive integer factors 2022?
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36. Answer 2-2
1352

37. Question 2-3

38. Question 2-3
Points M and N cut the line segment AB with A(-5, 6) and B(19, -12) into three equal parts such that M is closer to A. If C is the midpoint of AB, what is the length of MC?
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40. Answer 2-3 5

41. Question 2-4

42. Question 2-4
The equation 𝑥 2 + 𝑦 2 =2𝑥𝑦𝑧 has two solutions: (𝑎, 𝑎, 𝑎), and (𝑏, 𝑏, 𝑏), where a is distinct from b. What is 𝑎+𝑏?
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44. Answer 2-4 1

45. Question 2-5

46. Question 2-5
When Anna took her new car to get gas, the odometer read 347 mi. The next time she got gas it read 564 mi. How many palindromes had the odometer shown between the two fill-ups?
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48. Answer 2-5

49. Round 3

50. Question 3-1

51. Question 3-1
Find the two solutions to the following system. Give your answer as two ordered pairs.
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2nd Interval
1st Interval
5 Seconds
5 Seconds

53. Answer 3-1
(5, 25), (-1, 1)

54. Question 3-2

55. Question 3-2
Find the mean of the following values: the mean, median, and mode of the set
{1, 1, 2, 3, 5, 8, 1}.
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2nd Interval
1st Interval
5 Seconds
5 Seconds

57. Answer 3-2 2

58. Question 3-3

59. Question 3-3
Suppose a sphere with radius 𝑟 and a cube with a side length 𝑠 have the same volume.
If 𝑟 𝑠 =𝑥, find 𝑥 3 .
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2nd Interval
1st Interval
5 Seconds
5 Seconds

61. Answer 3-3

62. Question 3-4

63. Find the distance between the vertices of the two parabolas:
Question 3-4
Find the distance between the vertices of the two parabolas:
𝑦=−(𝑥−4 ) 2 −2
𝑦=2 𝑥 2 +12𝑥+21
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2nd Interval
1st Interval
5 Seconds
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65. Answer 3-4 𝟕𝟒

66. Question 3-5

67. Question 3-5 0:40 0:39 0:41 0:43 0:44 0:38 0:42 0:36 0:33 0:32 0:34
A square is drawn on a circle so that one vertex of the square is on the circle’s center, and the adjacent vertices are on the circle. If the circle’s radius is 3, what is the area of the shaded region in terms of 𝜋?
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2nd Interval
1st Interval
5 Seconds
5 Seconds

69. Answer 3-5
𝟗− 𝟗𝝅 𝟒

70. Round 4

71. Question 4-1

72. Question 4-1
The number 2019 can be written as in base 9 as 𝑎𝑏𝑐 𝑑 9 where a, b, c, and d represent the distinct digits of the number. What is 𝑏⋅𝑑+𝑎⋅𝑐 ?
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2nd Interval
1st Interval
5 Seconds
5 Seconds

74. Answer 4-1

75. Question 4-2

76. Question 4-2
Two numbers have an arithmetic mean of 5 and a geometric mean of 3. What is the sum of the square roots of the two numbers?
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2nd Interval
1st Interval
5 Seconds
5 Seconds

78. Answer 4-2

79. Question 4-3

80. Question 4-3
A triangle with vertices at (0, 8), (-3, 6), and (0, 0) has the same area as a square with side length x. Find 𝑥 4 .
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2nd Interval
1st Interval
5 Seconds
5 Seconds

82. Answer 4-3

83. Question 4-4

84. Question 4-4 Find the value of 1+2 1+2 1+2 1+... 0:41 0:42 0:40 0:44
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2nd Interval
1st Interval
5 Seconds
5 Seconds

86. Answer 4-4
𝟑+𝟐 𝟐

87. Question 4-5

88. Question 4-5 0:41 0:42 0:40 0:44 0:45 0:39 0:43 0:37 0:33 0:32 0:34
Riley the border collie is doing better than last year. The probability that Riley fetches the frisbee is now , and the probability that she fetches the ball is If the frisbee and the ball are each thrown twice, what is the probability that Riley fetches the ball on the first throw but not the second throw, then fetches the frisbee on the second throw but not the first throw?
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2nd Interval
1st Interval
5 Seconds
5 Seconds

90. Answer 4-5
𝟐𝟕 𝟏𝟐𝟐𝟓

91. Extra Questions

92. Question E-1

93. Question E-1
Find the equation of a line perpendicular to the line 𝑦= 3𝑥−4 that passes through the point (-5, 13). Give your answer in slope-intercept form.
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2nd Interval
1st Interval
5 Seconds
5 Seconds

95. Answer E-1
𝒚=− 𝟏 𝟑 𝒙+ 𝟑𝟒 𝟑

96. Question E-2

97. Question E-2
Two numbers have an arithmetic mean of 10 and a geometric mean of 8. What is the sum of the square roots of the two numbers?
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1:30
1:10
1:09
1:05
1:04
1:06
1:07
1:08
1:16
1:15
1:26
1:17
1:27
1:28
1:29
1:24
1:25
1:23
1:18
1:20
1:19
1:22
1:21
2nd Interval
1st Interval
5 Seconds
5 Seconds

99. Answer E-2 6

100. Question E-3

101. Question E-3 0:41 0:42 0:40 0:44 0:45 0:39 0:43 0:37 0:33 0:32 0:34
Amy lives five blocks south and four blocks east of the grocery store. She wants to walk there and back. If she can only travel north and west, how many paths could she take to get to the grocery store?
0:41
0:42
0:40
0:44
0:45
0:39
0:43
0:37
0:33
0:32
0:34
0:35
0:46
0:36
0:38
0:48
0:56
0:55
0:57
0:58
1:00
0:59
0:54
0:53
0:49
0:31
0:50
0:51
0:52
0:47
0:30
0:10
0:09
0:11
0:12
0:14
0:13
0:08
0:07
0:02
0:01
0:03
0:04
0:06
0:05
0:15
0:16
0:25
0:24
0:26
0:27
0:29
0:28
0:23
0:22
0:18
0:17
0:19
0:20
0:21
1:01
1:02
1:41
1:40
1:42
1:43
1:45
1:44
1:39
1:38
1:33
1:32
1:34
1:35
1:37
1:36
1:46
1:47
1:56
1:55
1:57
1:58
2:00
1:59
1:54
1:53
1:49
1:48
1:50
1:51
1:52
1:03
1:31
1:12
1:11
1:13
1:14
1:30
1:10
1:09
1:05
1:04
1:06
1:07
1:08
1:16
1:15
1:26
1:17
1:27
1:28
1:29
1:24
1:25
1:23
1:18
1:20
1:19
1:22
1:21
2nd Interval
1st Interval
5 Seconds
5 Seconds

103. Answer E-3
126