2006 Vanderbilt High School Mathematics Competition

1. 2006 Vanderbilt High School Mathematics Competition Ciphering Please send your first round cipherer to the front at this time

2. 2006 Vanderbilt High School Mathematics Competition Ciphering Guidelines Separate and completely fill out answer sheets Only answers written in the answer blank provided will be graded There will be two one-minute time frames; a correct answer in the first minute is worth 10 points and a correct answer in the second minute is worth 5 points. A 5-second warning will be announced before the end of each time frame. Please fold your answer sheet and hold it in the air during this warning to turn in your answer.

3. 2006 Vanderbilt High School Mathematics Competition Ciphering Guidelines (cont.) Answer sheets will only be accepted during the 5-second interval, and answer sheets raised after the end of the time frame will not be accepted . A student may not take his answer sheet back after a runner has taken it. You may submit only one answer sheet per question. As always, calculators and other forms of aid are prohibited and using them will result in immediate disqualification.

4. 2006 Vanderbilt High School Mathematics Competition Ciphering Guidelines (cont.) Do not approximate radicals or other irrational numbers such as Φ, π, and e unless specifically instructed otherwise in the problem. Fractions may be left in mixed (ex. 3 ½), improper (ex. 7/2), or decimal (ex. 3.5) form as long as they are fully reduced. For example, 14/4 would not be an acceptable answer.

5. ROUND 1 Practice Question

6. Practice Question A company’s employee identification numbers consist of 1 uppercase consonant, 1 lowercase vowel, and 3 odd digits. How many different identification numbers are possible? (Count the letter Y as a consonant)

7. ROUND 1 Question 1

8. Question 1.1 Urn I contains two black chips and one gold chip. Urn II contains one black chip and two gold chips. One chip is drawn from Urn I and transferred to Urn II. Then a chip is drawn from Urn II. Given that a black chip is drawn from Urn II, what is the probability that the transferred chip was black?

9. ROUND 1 Question 2

10. Question 1.2 In a plane, what is the set of all points equidistant from the set of all points equidistant from two perpendicular lines?

11. ROUND 1 Question 3

12. Question 1.3 An ant sitting in one corner of the front of a 3 x 4 x 5 closed box wants to walk to the opposite corner on the back of the box. What is the shortest walking distance?

13. ROUND 1 Question 4

14. Question 1.4 Dwarmby the clown has a bag with 4 red marbles, 4 blue marbles, 3 green marbles, and 3 magical invisible marbles. If Dwarmby picks 4 marbles at random, what fraction represents the probability that he will be able to see all the marbles he has picked?

15. ROUND 1 Question 5

16. Question 1.5 Gabriel is placing pennies on a chess board. She puts 2 on the first square, 4 on the second square, 8 on the third, 16 on the fourth, and so on. If the chessboard can only hold 16,800 pennies, how many squares can be filled before it collapses?

17. End of Round 1 Please send your next cipherer to the front to begin Round 2

18. ROUND 2 Question 1

19. Question 2.1 What is the remainder if 22006 is divided by 13?

20. ROUND 2 Question 2

21. Question 2.2 Evaluate the sum:

22. ROUND 2 Question 3

23. Question 2.3 A standard die has the 5 replaced with a 2. What is the expected sum of 2 rolls of the die?

24. ROUND 2 Question 4

25. Question 2.4 What is the sum of the cubes of the roots of 2x3 – 4x2 – 46x + 120 = 0?

26. ROUND 2 Question 5

27. Question 2.5 A triangle with base x has the same area as a rectangle whose height is 5 times that of the triangle. What is the ratio of the rectangle's width to the triangle's width?

28. End of Round 2 Please send your next cipherer to the front to begin Round 3

29. ROUND 3 Question 1

30. Question 3.1 Suppose that 2006 straight lines are drawn so that every pair of lines intersects but no three lines intersect at a common point. Find the sum of the digits in the number of regions into which these lines divide the plane.

31. ROUND 3 Question 2

32. Question 3.2 In how many distinct ways can you arrange the letters in the word “Rattler” if you count upper and lower case letters as distinct (i.e. R ≠ r)?

33. ROUND 3 Question 3

34. Question 3.3 How many times between 6 A.M. and 6 P.M. (of the same day) do the hands of a clock form the acute angle between the lines 3x + 4y = 7 and x + y = -6?

35. ROUND 3 Question 4

36. Question 3.4 What is the length of the period of

37. ROUND 3 Question 5

38. Question 3.5 The Men’s NCAA Basketball Tournament begins with 65 teams. After one “play-in game” that eliminates one team, the field is reduced to 64, at which point all 64 teams are paired and the losers of these matches are eliminated. This process of pairing and elimination is repeated for the remaining teams until 1 undefeated team remains. Find the sum of the positive integral divisors in X, if X is the total number of games needed to determine this champion.

39. End of Round 3 Please send your next cipherer to the front to begin Round 4

40. ROUND 4 Question 1

41. Question 4.1 77.89062510 = ______8 ?

42. ROUND 4 Question 2

43. Question 4.2 If Mark has to pay rent for his apartment once every minute, how many times did Mark pay rent between January 1, 1998 and December 31, 2001, inclusive?

44. ROUND 4 Question 3

45. Question 4.3 3 Commodores equal 7 Volunteers, 2 Bulldogs equal 3 Tigers, 5 Tigers equal 1 Gator, and 8 Bulldogs equal 13 Volunteers. How many Commodores equal 3 Gators?

46. ROUND 4 Question 4

47. Question 4.4 The product of n matrices has the form If the product is equal to Find n.

48. ROUND 4 Question 5

49. Question 4.5 Find the value of x2 + y2 + z2, where x, y, and z satisfy the following system: z + y – x = 8 2z – y = 3 y + x = 6

50. END OF CIPHERING Scores will be posted shortly