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Sprint TestRound 11713

Name:Grade:School:

place ID sticker inside this box

Score: #1 Scorer’s Initials


1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

Sprint Test - Round 11713 - c© 2016 mathleague.org


1. What is the value of 999× 2016?

2. The midpoints of the sides of a square are connected to form a smaller square. Whatis the ratio of the area of the smaller square to the larger square? Express your answeras a common fraction.

3. What is the largest quotient that can be formed by picking two numbers from the set−18,−10,−6,−2, 2, 4, 12, 16?

4. Lindsey purchased two Snickers for $1.25 each, and one bottle of water for $0.75. Howmany dollars change did Lindsey receive if she paid with a $5 bill? Express your answeras a decimal.

5. The digits 2, 3, 5, 7, and 8 are used to form a two-digit number and a three-digitnumber, using each digit only once. What is the smallest possible positive differencebetween the two-digit number and the three-digit number?

6. How many numbers are in the arithmetic sequence 2, 7, 12, 17, ..., 222?

7. At the beginning of a road trip, the odometer on Brandon’s car read 34,150 milesand his car had a full tank of gas. Brandon stopped when the odometer read 34,400and filled his tank with 8 gallons of gas. At the conclusion of the trip, the odometerread 34,654 miles and Brandon required 10 gallons of gas to fill his tank. What wasBrandon’s gas mileage for the trip, in miles per gallon?

8. Kylo purchased a Kyber crystal for his homemade lightsaber. Unfortunately, Kylocracked the crystal while installing it, and the crystal lost 35% of the value of itspurchase price. If the cracked crystal is now worth only 910 galactic credits, what wasthe crystal’s purchase price, in galactic credits?

9. What is the largest three digit number that has a remainder of 1 when divided by both9 and 7?

10. What is the sum of all positive integers that evenly divide 30?

11. A rectangle and a square have the same area. If one side of the square has a lengthof 12 and one side of the rectangle has a length of 16, what is the perimeter of therectangle?

12. At 5 o’clock, what is the degree measure of the smaller of the two angles between thehour and minute hands on a clock?

13. Eight breakfast tacos cost less than $9.00, but nine breakfast tacos cost more than$10.00. How much does one breakfast taco cost? Assume the price does not includefractional cents, and express your answer as a decimal.



14. How many integer values of x satisfy 2x2 − 5x− 7 ≤ 0?

15. If the sum of an integer and its square is 210, what is the least possible value of theinteger?

16. What is the product of all values of x that satisfy ||x− 2| − 3| = 5?

17. Mala and Joslyn number the cards in a standard deck of playing cards from 1 to52. Mala and Joslyn each draw a card and show them to each other. What is theprobability that the sum of the numbers on the cards is even? Express your answer asa common fraction.

18. A quadrilateral has vertices at (−8, 2), (−2,−3), (3, 3), and (−3, 8). What is the areaof the quadrilateral, in square units?

19. Suppose x and y are positive integers such that x3 − y2 = 343. What is the minimumpossible value of x+ y?

20. Circle O has radius 10. Chords AB and CD are parallel, separated by distance 10,and AB = CD. What is the area of the quadrilateral ABCD?

21. How many ordered triples of integers (x, y, z) satisfy 0 < x < y + 1 < z + 2 < 12?

22. How many lattice points lie on the circle defined by the equation x2 +y2−2x+4y = 8?

23. 4ABC has points D on AB and E on AC. Segment DB is 58

of AB, and segmentEC is 2

7of AC. What is the ratio of the area of 4ADE to the area of quadrilateral

DBCE?

24. Suppose a2− 9a+ 17 = 0 and b2− 9b+ 17 = 0, but a is not equal to b. What is ab

+ ba?

25. A semicircle with center O has radius 6. A and D are the endpoints of the semicircle’sdiameter, while B and C are points on arc AD so that AB = BC = CD. Computethe area of the region bounded by line segments BC, AC, and arc AB. Express youranswer in terms of π.

26. If the roots of the quadratic x2 + bx+ c− 10 = 0 are 7 and 14 for some integers b andc, what is the positive difference between the roots of the quadratic x2 + bx+ c = 0?

27. What is the quotient when 100000001000000013 is divided by 1000100013? Expressyour answer as a base-3 number.



28. Ian Bayes hosts a game show where contestants choose one of four doors. Behind oneof the doors is a new car, and behind three of the doors are goats. After the contestantschoose their door, Ian opens one of the three doors not chosen by the contestant toreveal a goat. Contestants may then decide to stick with their initial choice or switchto one of the two other doors that are still closed. The probability of winning the carif the contestant sticks with their first choice is P, and the probability of winning thecar if the contestant changes to one of the other two doors is Q. What is P

Q? Express

your answer as a common fraction.

29. What is the value of 11054+11053+2·11052+11061221026

?

30. a, b, and c are the lengths of the radii of three mutually externally tangent circles. Thearea of the triangle formed by connecting the centers of the three circles is 84, andabc = 252. What is the perimeter of the triangle formed by connecting the centers ofthe three circles?


1. A special deck of cards is comprised of cards with a letter printedon one side and a number printed on the other. Five cards are placedon the table, and the sides facing up show a D, an 8, a K, a 3, anda 2. Leo claims that every card that has a vowel on one side has aneven number on the other. Ray flips over one of the five cards cardand proves Leo wrong. Which symbol was on top of the card Rayflipped over?

1.

2. A cube five feet on a side has a three foot cube carved out fromthe center of one of its faces. By what percent does the surface areaof the original five foot cube increase? 2.

Target TestRound 11713

Name:

Grade:

School:




Target Test - Round 11713 - c© 2016 mathleague.org

3. Alan is a young entrepreneur. He buys pencils at the local Save-Mart for $3.00 for a package of one dozen. When a classmate needsa pencil in school, he happily sells them one for $0.50. How manypackages of pencils does Alan need to purchase if he wants to earna profit of $30.00?

3.

4. On her drive from Houston to Fredericksburg, Kara drove for onehour at 55 miles per hour and two hours at 70 miles per hour. Shethen had a flat tire and had to stop for half an hour to replace theflat with a spare. Because she couldn’t drive faster than 45 milesper hour on the spare, she drove the last 39 miles in one hour. Whatwas her average speed for the trip in miles per hour?

4.


Name:

Grade:

School:





5. When scoring a 30 question multiple choice test, 5 points areawarded for a correct answer, 3 points are deducted for an incorrectanswer, and 0 points are awarded if the answer is left blank. Annescored 108 on the test. How many questions did she leave blank?

5.

6. George writes down the numbers 2, 8, 12, and 14 on a sheet ofpaper. George’s brother Olen takes the sheet of paper and writesdown a fifth number, then remarks that the median of the five num-bers is equal to the mean of the five numbers. What is the sum ofall possible numbers that Olen could have written down?

6.


Name:

Grade:

School:





7. JB is an inexperienced mountain climber trying to climb Mt.Everest. From base camp, he makes it 1

2of the way to the peak on

the first day. On the second day, he makes it 13

of the remainder, andon the third day he makes it 1

4of the remainder. With resupplies

from his trusty Sherpa guide, this pattern continues as JB slowlyascends the world’s tallest mountain. On what day will JB be 90%of the way to the peak by the end of the day?

7.

8. How many ordered triples of integers (a, b, c) are solutions to theequation abc + ab + bc + ca + a + b + c = 1330?

8.


Name:

Grade:

School:





Team TestRound 11713

School/ Score: #1 Scorer’s InitialsTeam:


name or ID sticker goes in this box name or ID sticker goes in this box



1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Team Test - Round 11713 - c© 2016 mathleague.org


1. Define a#b as ab + a + b. What is the value of ((1#2)#3)#(4#5)?

2. GonzoBus provides bus service between Washington, DC and New York City, a tripthat takes 5 hours in either direction. Buses from Washington depart Union Stationevery hour at 40 minutes past the hour. Buses from New York City depart the PortAuthority Terminal every hour on the hour. Bryce is on a bus departing Washingtonat 8:40 AM. Assuming all buses take the same route, how many GonzoBuses will Brycesee traveling the other way on his journey to New York?

3. A square sheet of paper is folded in half twice in succession, with the direction of eachfold, horizontal or vertical, chosen at random. What is the ratio of the smallest possibleresulting perimeter to the largest possible resulting perimeter? Express your answeras a fraction.

4. There are forty-two teachers at Neil Armstrong Junior High School. Each teacherteaches five classes, and each class has twenty five students. There is exactly oneteacher per class, and each student is enrolled in six classes. How many students areenrolled at Neil Armstrong Junior High School?

5. A four digit number has a thousands digit that is even, a hundreds digit that is odd,and tens and units digits that are both different from the other three digits. How manysuch numbers are possible?

6. At Mary’s Marvelous Math Meet, students are seated in a large auditorium with fifteenrows of seats. The first row of seats in the auditorium has ten seats, and each additionalrow has two more seats than the previous row. If students seated in the same row musthave at least two empty seats between each other, what is the maximum number ofstudents that may be seated in the auditorium?

7. A lattice point is a point (x, y) where both x and y are integers. How many latticepoints are contained in the region defined by 0 ≤ x ≤ y ≤ 99?

8. Will picks a number between one and ten, inclusive, and takes the average of all positivemultiples of that number that are less than or equal to one thousand. Will’s brotherCharles does the same thing. If Will’s result was four more than Charles’ result, thenwhat was Will’s original number?

9. Consider the function f , defined as f(x, k) =

{2 · f(x− 1, k) + k if x > 0

1 if x = 0. What is

the value of f(5, 1) + f(5, 0) + f(5,−1)?



10. A regular tetrahedron of side length 2 is split into two congruent pieces by passinga plane through the center of the tetrahedron as well as the midpoints of four of theedges. What is the surface area of one of these pieces? Express your answer in simplestradical form.


CountdownRound 11713

1. Simplify: (5 +√

7)(5−√

7).

2. What is the area of an equilateral triangle with side length 2 · 37/4?

3. What is the units digit of 90092 − 1?

4. From a group of nine mathletes, Coach Z must select a team captain and three moreteam members. In how many ways can Coach Z select his team?

5. Round to the nearest tenth: 1918.3181

6. How many diagonals does a regular heptagon have?

7. Find the remainder when 7100 is divided by 2401.

8. How many prime numbers are between 50 and 100?

9. How many positive factors does 104 have?

10. Two chords, AB and CD, meet inside a circle at P. If AP = 11 and CP = 2, then whatis BP

DP? Express your answer as a common fraction.

11. If two positive integers have a sum of 44, what is their maximum possible difference?

12. Given xy

= 611

and yz

= 223

, find xz.

13. Bob walks 3 blocks north, 5 blocks east, 9 blocks south, and 13 blocks west. How manyblocks is he from his starting point?

14. Compute the remainder when 17768 is divided by 7.

15. What is the greatest integer less than 3√−121?

16. Express as a common fraction: 15

+ 152

+ 153

+ . . . .

17. How many ordered pairs of integers (x, y) satisfy 1x

+ 1y

= 13?

18. A baseball team lost 37.5% of the 120 games they played. How many games did theywin?

19. What is the area of the triangle that has vertices at (0, 0), (10, 0), and (12, 12)?

20. A quadrilateral has its vertices chosen from a 9 by 9 grid of points. What is the areaof the largest possible quadrilateral that satisfies this condition?

21. If x, y, z are distinct integers such that xyz = 24, then what is the minimum possiblevalue of x+ y + z?

Countdown - Round 11713 - c© 2016 mathleague.org


22. The average of six consecutive integers is 8.5. What is the smallest of the six integers?

23. A certain species of bacteria doubles in population every 8 hours. If a container startsoff with one bacterium, how many whole days would it take for the total number ofbacteria to exceed a quarter million?

24. What is the largest prime number less than 600?

25. A circular cylinder has radius 3 and height 8. Express its surface area in terms of π.

26. What is the maximum number of intersections of 6 lines if 3 of them are parallel?

27. Calculate the volume of a right circular cylinder with base area 26π and height 5.

28. Kevin has a drawer of socks. Each sock is one of four sizes and one of two colors. Howmany socks must he select from the drawer to ensure he has at least one matching pairof socks? A matching pair of socks is one in which both socks are the same size andcolor.

29. The ratio between two consecutive even integers is 910

. Find the sum of the integers.

30. Calculate: 912.

31. If 2x2y = 90, and x = 3, then what is the value of x6y2?

32. If x− y = xy = 3, what is x3 − y3?

33. Simplify: 2 + 34 − 5× 6.

34. The measures of the angles of a triangle are in the ratio 3 : 4 : 5. Find the number ofdegrees in the positive difference between the largest and the smallest of the angles.

35. What is the maximum number of regions three lines can divide the plane into?

36. The width of a rectangle is increased by 100%. If the area remains constant, by whatpercent must the length of the rectangle decrease?

37. How many 4-person committees can be formed from a group of 7 people?

38. Find the sum of the positive integer divisors of 42.

39. How many positive integers less than 25 are not factors of 9! ?

40. What is the sum of first 49 positive integers?

41. What is the volume of a cube with side length 12?



42. What number is 56% of 75?

43. How many 2-digit numbers are equal to 5 times the sum of their digits?

44. The area of a right triangle with integer side lengths is 30. What is its perimeter?

45. Express as a decimal: 9512.5

.

46. In a class of 28 students, there are 20 students who play a sport and 18 who play aninstrument. What is the minimum number of students who play both a sport and aninstrument?

47. How many of the first 100 positive integers are divisible by at least one of 2 and 5?

48. There are 52 passengers on a bus. At every odd numbered stop 9 passengers exit thebus and at every even numbered stop 2 passengers board the bus. The bus startsat stop 1, where the first group of people get off. Find the first stop where the busbecomes empty.

49. What is the largest integer n such that 2n divides 384?

50. What is the slope of the line x = 9y + 9? Express your answer as a common fraction.

51. How many odd three-digit positive integers have no digits greater than 4?

52. If pens cost $0.30 each or 4 for $1.00, how much is saved on the purchase of two dozenpens by buying them in sets of 4?

53. If 2 sides of a right triangle have length 6 and 8, what is the least possible value of thethird side? Express your answer in simplest radical form.

54. How many digits are required to write 2.718 · 1099 in decimal notation?

55. What is the largest prime factor of 2499?

56. Express as a mixed number: 313

+ 515

+ 15 115

.

57. A cube has the volume of 512. What is its total surface area?

58. What is the sum of the positive multiples of 13 less than 175?

59. What is the minimum value of x2 − 6x+ 19?

60. If the measure of one angle of a quadrilateral is 80 degrees and all of the angles forman arithmetic sequence, what is the largest possible measure of the largest angle in thequadrilateral?



61. How many perfect squares are factors of 288?

62. If the length of a rectangle is 2 feet less than twice times its width and its perimeteris 38 feet, what is the area of the rectangle?

63. If the lengths of the axes of an ellipse are 4 and 10, what is the area of the ellipse?Express your answer in terms of π.

64. Find the next number in the pattern: 2, 4, 7, 11, 16, 22, .

65. Two roots of the equation 2x3 − 8x2 + px+ q = 0 are 5 and 3. Find the third root ofthe equation.

66. If one-sixteenth of 22020 is equal to 2x, what is x?

67. Compute: (−2)4 · 2−3.

68. Compute the sum of the roots of x3 + 15x+ 99.

69. If x+ y = 55, x+ z = 65, and y + z = 80, what is the value of x+ y + z?

70. Ten coins, all quarters and dimes, are worth a total of $2.05. What percent of thecoins are dimes?

71. How many ways can you make 26 cents with pennies, nickels, dimes, and quarters?

72. The circumradius of a triangle with side lengths 34, 288, and 290 is 145. Find thedegree measure of the largest angle of the triangle.

73. Convert 1001011102 to base 8.

74. Find the remainder when 267498 is divided by 9.

75. Compute the sum:(60

)+(51

)+(42

)+(33

).

76. Byron runs at 8 m/s while Ron runs at 7 m/s. They both begin running around a400 meter track in the same direction at the same time. After how many minutes doesByron pass Ron for the sixth time?

77. Billie drives at 30 miles/hour for 1 hour and at 60 miles/hour for 2 hours. What is hisaverage speed, in miles per hour, for the three hours?

78. Evaluate: 2 + 4 + 6 + · · ·+ 28.

79. There exists an x such that y = 5x− 1 and y = −10x+ 101. Find y.

80. How many cups are there in 2 gallons?


SolutionsRound 11713

Sprint Test

1. 2013984

2. 12

3. 9

4. ($) 1.75

5. 148

6. 45

7. 28

8. 1400

9. 946

10. 72

11. 50

12. 150(◦)

13. ($) 1.12

14. 5

15. −15

16. −60

17. 2551

18. 61

19. 21

20. 100√

3

21. 165

22. 8

23. 1541

24. 4717

25. 6π

26. 3

27. 222200013 or 22220001

28. 23

29. 1222131

30. 56

Target Test

1. 3

2. 24(%)

3. 10

4. 52

5. 2

6. 37

7. 9

8. 40

Team Test

1. 719

2. 10

3. 45

4. 875

5. 1120

6. 125

7. 5050

8. 10

9. 96

10. 2√

3 + 1

Solutions - Round 11713 - c© 2016 mathleague.org


Countdown

1. 18

2. 81

3. 0

4. 504

5. 1918.3

6. 14

7. 0

8. 10

9. 8

10. 211

11. 42

12. 4

13. 10

14. 0

15. -5

16. 14

17. 5

18. 75

19. 60

20. 64

21. -24

22. 6

23. 6

24. 599

25. 66π

26. 12

27. 130π

28. 9

29. 38

30. 8281

31. 18225

32. 54

33. 53

34. 30

35. 7

36. 50

37. 35

38. 96

39. 6

40. 1225

41. 1728

42. 42

43. 1

44. 30

45. 7.6

46. 10

47. 60

48. 15

49. 7

50. 19

51. 40

52. $1.20

53. 2√

7

54. 100

55. 17

56. 23 35

57. 384

58. 1183

59. 10

60. 120

61. 6

62. 84

63. 10π

64. 29

65. -4

66. 2016

67. 2

68. 0

69. 100

70. 30%

71. 13

72. 90

73. 4568

74. 0

75. 13

76. 40

77. 50

78. 210

79. 33

80. 32



Sprint Test Solutions

1. We could multiply it out, or we could note that999·2016 = 2016·(1000−1) = 2016000−2016 =2013984.

2. Let the side of the larger square be s, givingthe larger square an area of s2. The smallersquare has a side that is the hypotenuse of anisosceles right triangle with legs of s

2 . By the

Pythagorean Theorem, that hypotenuse is s√2

2 ,and the area of the smaller square is therefores2

2 . The ratio of the area of the smaller square

to the larger square iss2

2

s2 = 12 . Alternatively,

we can see that the corners of the larger squarecan be folded over to exactly cover the smallersquare, meaning that the smaller square is ex-actly half of the larger square.

3. To form the largest quotient, both the numer-ator and denominator chosen must be greaterthan 0 or less than 0, since otherwise the quo-tient would be negative. Furthermore, we mustmaximize the absolute value of the numeratorand minimize the absolute value of the denom-inator in each case. For the case where bothare positive, the largest quotient is 16

2 = 8. Forthe case where both are negative, the largestquotient is −18−2 = 9. Thus the answer is 9.

4. Lindsey’s total purchase is 2 · $1.25 + $0.75 =$3.25. The change received is $5.00 − $3.25 =$1.75.

5. To minimize the difference, we want the three-digit number to be as small as possible andthe two-digit number to be as large as possible.Thus, we have a smallest possible difference of235− 87 = 148.

6. The terms of the sequence increase arithmeti-cally, with a common difference of 5. The nthterm can therefore be written as 5n − 3. So222 = 5n− 3 and n = 45.

7. Brandon traveled 34654 − 34150 = 504 miles,and he used 8 + 10 = 18 gallons of gas. His gasmileage was 504

18 = 28 miles per gallon.

8. If the crystal lost 35% of its value, then it re-tained 65% of its value. Therefore the purchaseprice was 910

0.65 = 1400 galactic credits.

9. If a number has a remainder of 1 when dividedby 9 and 7, then it has a remainder of 1 whendivided by the least common multiple of 9 and7, which is 63. The largest three digit multipleof 63 is 15 · 63 = 945, so the largest three digitnumber that has a remainder of 1 when dividedby 63 is 946.

10. We make a list of the factors of 30:1, 2, 3, 5, 6, 10, 15, 30. Their sum is 1 + 2 + 3 +5+6+10+15+30 = 72. Also, since 30 = 2·3·5,the factors can be generated by the expansionof (1 + 2)(1 + 3)(1 + 5), and so the sum mustbe 3 · 4 · 6 = 72.

11. The area of the square is 122 = 144. The otherleg of the rectangle is 144

16 = 9. The perimeterof the rectangle is 2 · (16 + 9) = 50.

12. At 5 o’clock, the minute hand points exactlyat the 12 and the hour hand points exactly atthe 5. Each hour mark on the clock face is360◦

12 = 30◦. Therefore the degree difference be-tween the 12 and the 5 is 5 · 30◦ = 150◦.

13. Let p be the cost of a breakfast taco. We have9p > 10 and 8p < 9. Therefore p > 1.1 andp < 1.125. The only price that satisfies bothinequalities is $1.12.

14. Factoring, we have (x+ 1)(2x− 7) ≤ 0. There-fore x ≥ −1, and x ≤ 3.5, so the inequality issatisfied by −1, 0, 1, 2, and 3, for a total of 5values.

15. We have x2 +x = 210. Rearranging and factor-ing, (x + 15)(x − 14) = 0, and so the smallestpossible value of x is −15.

16. Clearing the outer set of absolute value signs,we get |x− 2| − 3 = ±5. But |x− 2| − 3 clearlycannot equal -5, so |x − 2| = 8, and x = 10 orx = −6.

17. The sum can only be even if both cards are



even or both cards are odd. Once Mala drawsher card, Joslyn must choose one of the 25cards that matches the parity (the odd/evenproperty) of Mala’s card from the remaining 51cards. Thus the probability of an even sum is2551 .

18. Plotting the points, one can see that they forma square with side length

√62 + 52 =

√61, and

so the quadrilateral has an area of 61. (One canverify that this is a square by checking that allsides have the same length, and two sides areperpendicular to each other.)

19. Rearranging gives x3 − 343 = x3 − 73 = (x −7)(x2 + 7x + 49) = (x − 7)((x − 7)2 + 21x). Ifz = x − 7, then we want z(z2 + 21z + 147) tobe a perfect square. Miraculously, the expres-sion evaluates to 169 = 132 at z = 1, so x = 8and x + y = 21. Alternatively, note that thesmallest possible value of x is 8, and that workswith y = 13. This must yield the smallest x+ybecause if x increases, y must increase as well.

20. Because AB and CD are parallel and of equallength, they must be equidistant from the cen-ter of O, and ABCD is a rectangle. The diago-nal of the rectangle is 20, and one side is 10. Bythe Pythagorean theorem, the other side of therectangle is

√202 − 102 = 10

√3, and the area

of the rectangle is 10 · 10√

3 = 100√

3.

21. We can pick the set {x, y + 1, z + 2} from theset {1, 2, . . . , 11} in

(113

)= 11·10·9

3·2 = 165 ways,so there are 165 triples (x, y, z).

22. Completing the squares, we have (x− 1)2− 1 +(y + 2)2 − 4 = 8, and (x − 1)2 + (y + 2)2 =13. That equation defines a circle with cen-ter (1,−2), and radius

√13. Therefore we need

pairs of squares which sum to 13. The only suchpairs are ±2 and ±3, so the 8 lattice points de-fined by (±2,±3) and (±3,±2) lie on the circle.

23. Since segment BD is 58 of AB, AD is 3

8 ofAB. Similarly, AE is 5

7 of AC. Let the area of4ABC be 1. Because ∠A is common to bothtriangles, the ratio of the areas of 4ADE to

4ABC is ADAB ·

AEAC , which is 3

8 ·57 = 15

56 . Sincewe said the area of4ABC is 1, 15

56 is the area of4ADE, and the area of DBCE is 1− 15

56 = 4156 .

Thus the ratio of the area of4ADE to the areaof DBCE is 15

41 .

24. We have ab + b

a = a2+b2

ab = (a+b)2−2abab = (a+b)2

ab −2. Since a and b are the roots of the polynomialx2−9x+17, we have that a+b = 9 and ab = 17,so the answer is 81

17 − 2 = 4717 .

25. Draw in segments AC and BO, and let P betheir intersection. Note that AB = BC =CD = 6, and ABCO is a parallelogram. Then4BPC and4OPA are congruent. The desiredarea is the combined areas of 4BPC, 4APB,and the region between segment AB and arcAB. Replacing 4BPC with 4OPA, we arenow looking for the area of sector OAB, whichis 1

3 of the original semicircle. This area is 6π.

26. By Vieta’s formulas, b = −(7 + 14) = −21 andc − 10 = 7 · 14 = 98, so c = 108. Therefore,we want the quadratic x2 − 21x + 108 = 0, or(x − 9)(x − 12) = 0, so the difference betweenthe roots is |12− 9| = 3.

27. Note that this is equal to the base-3 repre-

sentation of 316+38+138+34+1 . Let x = 34. Then

our expression is x4+x2+1x2+x+1 , which is equal to

x2 − x + 1 = 38 − 34 + 1. The base-3 repre-sentation of this is 222200013.

28. The probability of initially choosing the doorwith the car is 1

4 . That probability does notchange when Ian opens one of the other threedoors, which he will always be able to do sinceat least two of them must reveal goats. There-fore P = 1

4 and the probability of the car beingbehind one of the other two doors is 1−P = 3

4 .

Thus Q = 38 , and P

Q =1438

= 23 .

29. The given expression may be rewritten as11054+11053+2·11052+1105+1

11052+1 . For brevity, letx = 1105. The expression then becomesx4+x3+2x2+x+1

x2+1 , which after performing poly-



nomial long division (or factoring) reduces tox2 + x+ 1 = 11052 + 1105 + 1 = 1222131.

30. The perimeter of the triangle formed by con-necting the centers of the three circles is (a +b) + (b+ c) + (a+ c) = 2a+ 2b+ 2c. By Heron’s

formula, the area of a triangle with side lengthsx, y, z is

√s(s− x)(s− y)(s− z). Plugging in

what we know gives us√

(a+ b+ c)abc. We

know√

252(a+ b+ c) = 84, meaning a+b+c =28 and the perimeter is 56.

Target Test Solutions

1. Leo can be proven wrong by showing that thereis a card that has a vowel on one side and anodd number on the other. Thus Ray must haverevealed a vowel after flipping over the 3.

2. The initial surface area is ·52 = 150. When thethree foot cube is carved out, the net changeto the surface area of the object is the additionof the four vertical faces of the smaller cube,which have a total surface area of ·32 = 36.The percent increase is 36

150 = 24%.

3. When Alan sells a dozen pencils, he earns aprofit of 12·$0.50−$3.00 = $3.00. Thus to earna profit of $30.00, he must purchase $30.00

$3.00 = 10packages.

4. Average speed is total distance divided by to-tal time. The total distance Kara traveled is55 ·1+70 ·2+39 ·1 = 234 miles. The total timefor the trip is 2 + 1 + 0.5 + 1 = 4.5 hours. Thusthe average speed is 234

4.5 = 4689 = 52 miles per

hour.

5. Observe that an equivalent set of scoring rulesis 5 points for any question answered, correct orincorrect, with a loss of 8 points for an incorrectanswer. Letting A be the number of questionsanswered and W be the number answered in-correctly, we have 5A − 8W = 108. BecauseA < 30, the only valid solution is A = 28 andW = 4. The number left blank is 30−A = 2.

6. Let n be the number Olen wrote down. Themedian of the five numbers must be 12, 8, or n.The mean of the five numbers is 2+8+12+14+n

5 =36+n

5 . If the median is 8, then we need 36+n5 =

8, for which n = 4. If the median is 12, then36+n

5 = 12, for which n = 24. Finally, if themedian is n, then 36+n

5 = n, for which n = 9.The sum is 4 + 24 + 9 = 37.

7. After the first day, 12 of the journey remains.

After the second day, 12 ·

23 = 1

3 remains. Andafter the third day, 1

2 ·23 ·

34 = 1

4 remains. Thispattern continues, and on the end of day n, 1

n+1remains. Note that 90% completed is equiva-lent to 10% remaining. This occurs when ourproduct is 1

10 , which is at the end of day 9.

8. Adding 1 to both sides, we can factor the left-hand side to get (a + 1)(b + 1)(c + 1) = 1331.Note that 1331 = 113. Therefore, we wantto find the number of ways to write 113 asa product of three integers. Ignoring orderand signs, we have the solutions 113, 110, 110,112, 111, 110, and 111, 111, 111. The first onecan be ordered in 3 ways, the second one canbe ordered in 6 ways, and the third can be or-dered in 1 way. Thus, ignoring signs, there are3+6+1 = 10 solutions. For any solution, eithera + 1, b + 1, and c + 1 are all positive, or twoof them are negative; thus, there are 1 + 3 = 4ways to pick the signs for a+1, b+1, and c+1.Therefore, there are 4 · 10 = 40 triples total.

Team Test Solutions

1. Simplifying from the inside out, 1#2 = 5,5#3 = 23, and 4#5 = 29. Finally, 23#29 =719.

2. Ten minutes into his trip, it is 8:50 AM, and

being ten minutes from Union Station, Brycesees the bus that left New York 4 hours and 50minutes earlier, at 4:00 AM. Bryce will arrivein New York at 1:40 PM. In that time he willsee all buses that departed between 4:00 AMand 1:00 PM, inclusive, which is 10.



3. Let the length of the side of the square piece ofpaper be s. If the folds are in opposite direc-tions, the result will be a square of side s

2 , for aperimeter of 2s. If the folds are in the same di-rection, the result will be a rectangle with sidess and s

4 , for a perimeter of 5s2 . The ratio of

the smaller perimeter to the larger perimeter is2s5s2

= 45 .

4. With 42 teachers teaching 5 classes each, andexactly 1 teacher per class, there are 42·5 = 210classes being taught. Each class has 25 stu-dents, for a total of 25 · 210 = 5250 class enroll-ments. But each student is enrolled in 6 classes,so there must be 5250

6 = 875 students.

5. There are 4 choices for the thousands digit, 5choices for the hundreds digit, 8 choices for thetens digit, and 7 choices for the ones digit. Alto-gether, there are 4 ·5 ·8 ·7 = 1120 such numbers.

6. Let n be the number of seats on a row. Weobserve that if n has a remainder of 1 when di-vided by 3, then the row may contain bn3 c + 1students, by seating students on each end ofthe row and every third seat in between. Thusfor the front row, with ten seats, we can seat4 students. Adding two seats does not allowus to add an additional student, so if n hasa remainder of 0 when divided by 3, we canstill only seat n

3 students. Similarly, if n hasa remainder of 2 when divided by 3, then therow may still only contain bn3 c + 1 students.We now consider the number of seats in eachrow, the sequence 10, 12, 14, 16, ...38. We par-tition this into three sequences based on eachterm’s remainder when divided by three, givingus {10, 16, ...34}, {12, 18, ...36}, {14, 20, ...38}.The number of students becomes the sum 2 ·(4+6+8+10+12)+(5+7+9+11+13) = 125.

7. Consider a simpler case. For 0 ≤ x ≤ y ≤ 0,only the point (0, 0) works. For 0 ≤ x ≤ y ≤ 1,

only the three points (0, 0), (0, 1), and (1, 1)satisfy. Continuing on, we see that as the up-per bound increases from n − 1 to n, we addn + 1 new lattice points to the region. So theregion defined by 0 ≤ x ≤ y ≤ n contains1+2+3+...+n+1 points. Thus we need the sum

1+2+3+ · · ·+100, which is 100·(100+1)2 = 5050.

8. Let the number Will or Charles picks be n.We are looking for the average of the sequencen, 2n, 3n, . . . , kn, where k = b 1000n c. We make alist of pairs (n, kn), which is (1, 1000), (2, 1000),(3, 999), (4, 1000), (5, 1000), (6, 996), (7, 994),(8, 1000), (9, 999), (10, 1000). The average ofeach sequence is simply the average of the firstand last terms of the sequence. Those aver-ages, in order, are 500.5, 501, 501, 502, 502.5,501, 500.5, 504, 504, and 505. The largest av-erage is 505, the smallest is 500.5, and no aver-age is 504.5. For Will’s result to be four morethan Charles’, he must have picked the numberwhich resulted in an average of 505, which is10.

9. Since the function is only recursive on x, weevaluate the function for different values of k.When k = −1, we note that f(x,−1) = 1for all non-negative integer values of x. Whenk = 0, f(x, 0) = 2x for all non-negative inte-ger values of x. When k = 1, testing a fewvalues shows the pattern f(x, 1) = 2x+1 − 1for all non-negative integer values of x. Thusf(5, 1)+f(5, 0)+f(5,−1) = 26−1+25+1 = 96.

10. The total surface area of the two figures will bethe surface area of the tetrahedron plus twicethe area of the cross section. The cross sectionwill be a square with side length 1. Each faceof the tetrahedron has area

√3, so total sur-

face area is 4√

3, half of which goes to each ofthe two congruent pieces. Add the area of thesquare cross section (1), and we get a surfacearea of 2

√3 + 1.


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