2007 Vanderbilt High School Mathematics Competition Varsity Ciphering Please send your first round...

2007 Vanderbilt High School Mathematics Competition

Varsity Ciphering

Please send your first round cipherer to the front at this time

2007 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.

2007 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.

2007 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.

Round 1

Practice Question

Practice Question

What is the area of an isosceles trapezoid whose

area is 4?

Round 1

Question 1

Question 1.1

Find the sum of the entries in

1

14

73

Round 1

Question 2

Question 1.2

What is the distance between the graph of

4x + 3y – 7z = 0

and the point (1,6,2)?

Round 1

Question 3

Question 1.3

Find31272319

30262218

29252117

28242016

Round 1

Question 4

Question 1.4

What is the sum of the coefficients of

(2x + 3y + 4w – 6z)5 ?

End of Round 1

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Round 2

Question 1

Question 2.1

Find the sum of the coefficients of the first three terms in the expansion of

(A + 3B)7

Round 2

Question 2

Question 2.2

The two shortest sides of a triangle measure 39 and 80. If

one angle is α and another angle β is the complement of

α, find the length of the longest side.

Round 2

Question 3

Question 2.3

On a ten-question true/false quiz, what is the probability

that Archie gets 70% or above if he guesses on

every question?

Round 2

Question 4

Question 2.4

3 girls—Alley, Laura, and Molly—and 3 boys—Sam, Phil, and Chris—decide to be spontaneous and drive to Memphis at midnight. There are only 5 seats in the car. If a boy has to drive and another boy must ride in the front passenger seat and Sam must sit in the left window seat of the backseat, how many ways are there to arrange the six people?

End of Round 2

Please send your next cipherer forward at this time

Round 3

Question 1

Question 3.1

P(B|A) = .45

P(A)/P(A’) = 0.66666…

P(B’|A’) = 0.1

P(B’) = ?

Round 3

Question 2

Question 3.2

Find

n

n n

67

1lim

Round 3

Question 3

Question 3.3

Find !

8

!73 i

Round 3

Question 4

Question 3.4

What is the sum of the 4th triangular number, the 3rd

rectangular number, and the 2nd pentagonal number?

End of Round 3

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Round 4

Question 1

Question 4.1

tan(75°) tan(30°) tan(45°) tan(120°) = ?

Round 4

Question 2

Question 4.2

Find 729log3

2)2(log4 1818

Round 4

Question 3

Question 4.3

Simplify:

11

11

11

11

xy

yx

yx

yx

Round 4

Question 4

Question 4.4

For the shape defined by x2 + y2 – 12x– 14y – 43 = 0: A = the x coordinate of the center B = the y coordinate of the center C = ½(the longest possible distance of a line running through the center with endpoints on the shape) D = the eccentricity A + B + C – D = ?

End of Ciphering

Extra Question # 1

Question E.1

Adam Ant is crawling from the bottom left of a soda can to the top right. What is the length of the shortest possible path he can take if the height of the can is 4 in. and the radius of the base of the can is 2 in?

Extra Question # 2

Question E.2

If A = The measure in degrees of an interior angle of a regular 17-gon B = The sum of the roots of -2x2 – 16x + 8 = 0 C = The product of the roots of the equation in B D = The measure in degrees of an exterior angle

of a regular 17-gon

Find -2B + A + B + C + D + (B – C)