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 F, p, 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 companys 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?

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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?

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End of Round 2

• Please send your next cipherer to the front to
  begin Round 3

29
ROUND 3

• Question 1

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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.

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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)?

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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?

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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 Mens 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.

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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?

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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

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END OF CIPHERING

• Scores will be posted shortly

51
EXTRA QUESTION 1
52
Extra Question 1

• How many consecutive zeros does
• have at the end?

53
EXTRA QUESTION 2
54
Extra Question 2

• How many different sets of numbers
  s1,s2,,s5 consisting of only -1,0, and 1 are
  there such that s1s2 s3s4s5 -1?