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MC 2000 State Round Review

MC 2000 State Round Review. Let X be the size of the original. After two reduction, the new size is 2/3 * 2/3 X = 4/9 X. To turn it back to X, it needs to multiple by 9/4 = 225%. Answer: 225. 12. 9. 9. 3 + 4 + 3 + 6 = 16 . H. Per share measure = 48/16 = 3. 18.

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MC 2000 State Round Review

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  1. MC 2000 State RoundReview

  2. Let X be the size of the original. After two reduction, the new size is 2/3 * 2/3 X = 4/9 X To turn it back to X, it needs to multiple by 9/4 = 225% Answer: 225

  3. 12 9 9 3 + 4 + 3 + 6 = 16 H Per share measure = 48/16 = 3 18 Each size measure: 3*3 = 9, 4*3 = 12, 3*3 = 9, and 6*3 = 18 Assume H is the height of the trapezoid. H2 = 92 – ((18 – 12)/2)2 = 81 – 9 = 72  H = 62 Area = (12 + 18)/2 * H = 15 * 62 = 902

  4. a b d e c f g h Number of  with 1 block: 8 {a, b, c, d, e, f, g, h} Number of  with 2 blocks: 5 {ab, ad, be, de, gh} Number of  with 3 blocks: 6 {cde, def, cgh, ghf, adc, bef} Number of  with 4 blocks: 2 {acdg, befh} Number of  with 5 blocks: 2 {abcde, abdef} Number of  with 6 blocks: 0 Number of  with 7 blocks: 0 Number of  with 8 blocks: 1 {abcdefgh} Total = 8 + 5 + 6 + 2 + 2 + 1 = 24

  5. The only ones that can’t be reached with 3 scores (under 30)include: 19, 22, 25, 26, 28, 29 Answer: 6

  6. Note that there are 3 kinds of cards. If we pick 4 cards, two cards must have the same color Similarly, if we pick 7 cards, 3 of the them must have the same color Answer: 7

  7. Start with M, there is 1 choice There are 4 A’s surrounding M. WLG, Pick the green A. There are 3 T’s surround A. The T on top has 3 choices of H, and the Ts on the left/right have 2 choices of H Hence total # of choice = 1 * 4 * (1 * 3 + 2 * 2) = 4 * 7 = 28

  8. Drawing Venn Diagram as left 109 117 70 # of student in circles = 400 – 164 = 236 34 29 Let X be the ones take all three courses 114 236 = 117 + 109 + 114 – 70 – 34 – 29 + X X = 29

  9. Prime numbers below 24 (in increasing order): 2, 3, 5, 7, 9, 11, 13, 17, 19, 23 Set of 3 primes whose sum is 24 include: {2, 3, 19} {2, 5, 17} {2, 11, 11} {2, 3, 19}, {2,5, 17} each can form 6 ordered triplets {2, 11, 11} can form 3 ordered triplets Total # of triplets: 6 + 6 + 3 = 15

  10. Total size = 3X + 62 + 7X + 30 + 5X + 50 = 15X + 142 The perimeter is smallest if X is smallest If 3X + 62 = 7X + 30  4X = 32  X = 8 If 3X + 62 = 5X + 50  2X = 12  X = 6 If 7X + 30 = 5X + 50  2X = 20  X = 10 Smallest X = 6 Answer: 15 * 6 + 142 = 90 + 142 = 232

  11. Probability to draw a blue = 1 – 2/5 – 3/7 = 6/35 # of blue marbles is an integer, and the total # of marbles < 50 Total # of marbles = 35 # of blue marbles = 6 Prob of drawing 2 consequtive blue marbles = 6/35 * 5/34 = 3/119

  12. Note that the largest digit is 9. Three digits sum to 26  there are two 9s and one 8. By 899 * 7 * 11 * 13 = 899899 Answer: 4

  13. Let X be Joe’s price per pound, Y be # of pounds Joe bought X * Y = 352 ----------- (1) (X – 10) * (20 – Y) = 48 ---------- (2) From (2): 20 X + 10 Y - XY - 200 = 48 ---------- (3) (3) + (1): 20 X + 10Y – 200 = 400 Y = 60 – 2X --------- (4) (4) into (1): 60X - 2X2= 352  X2 - 30X + 176 = 0 (X – 22) (X – 8) = 0  X = 8 or 22 From (4), Y = 44 or 16; but 44 > 20 not valid  Answer: 16

  14. # of arrangement of any 3 elements: 3 * 2 * 1 = 6 Since a < b <c is one of the 6 arrangements Answer: 1/6

  15. Total number of odd multiple 3’s: 92 div 6 = 15 Total number of odd multiple 5’s: 92 div 10 = 9 Total number of odd multiple 3x5’s: 92 div 30 = 3 Sum of 3’s = 15 * (3 + 87) / 2 = 675 Sum of 5’s = 9 * (5 + 85) / 2 = 405 Sum of 15’s = 3 * (15 + 75) / 2 = 135 Sum of M = 675 + 405 – 135 = 945

  16. Let M & N be the # of sizes of the regular polygons Total # of interior degrees = (N-2)*180 + (M-2)*180 = 1980 (N – 2) + (M – 2) = 11  N+M=15  N = 15 – M --------- (1) Total # of diagonals = (N-3)N/2 + (M-3)M/2 = 34 N2 – 3N + M2 – 3M = 68 ---------- (2) From (1): 225 – 30M + M2 – 45 + 3M + M2 - 3M = 68 From (1): 2M2 – 30M +112 = 0 M2 – 15M + 56 = 0  (M – 7)(M – 8) = 0 M = 7 or M = 8  N = 8 or N = 7 Answer: 8 – 7 = 1

  17. # of ways to pick 2-buttons: 5 * 4 / 2 = 10 (note that sequence doesn’t matter!) # of ways to pick the next button: 5 Total # of combinations = 10 * 5 = 50

  18. R R Let R be the radius of the circle. Area(green-triangle) = 3/4 R2 Note that the height of the red-triangle is also R Area(red-triangle) = ½ * R * 2*(1/3)R = 1/3 R2 Area(small-hexagon)/Area(larger-hexagon = 6 * Area(green-triangle) / (6 * Area(red-triangle) ) = (3/4 R2) / (1/3 R2) = 33/4 = 3/4

  19. (0,6) Rotation around X-axis, form 2 cones. With base-area 62 , and height 5 (-5,0) (5,0) Volume (solid around X-axis) = (1/3 * 62  * 5 ) * 2 = 120  Rotation around Y-axis, form 1 cone. With base-area 52 , and height 6 Volume (solid around Y-axis) = (1/3 * 52  * 6 ) = 50  Answer: 120 – 50 = 70

  20. Notes on every quarter-hour: 4 Notes on every half-hour: 8 Notes on every 3-quarter-hour: 12 Notes on every hour: 16 Total in 24-hours = (4 + 8 + 12 + 16) * 24 = 40 * 24 = 960 Additional notes on the hour in 24 hours = (1 + 2 + … + 12) * 2 = 13*12 = 156 Answer = 960 + 156 = 1116

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