2007 Vanderbilt High School Mathematics Competition Junior 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

What is a billion plus two?

Round 1 Question 1

Question 1.1 Find

Round 1 Question 2

Question 1.2 A circle is inscribed inside a triangle with sides of lengths 4, 6, and 8. Find the square root of the ratio of the area of the triangle to the radius of the circle.

Round 1 Question 3

Question 1.3 If 35° is added to an angle, the resulting angle is equal to the supplement of the original angle. Find the complement of the original angle.

Round 1 Question 4

Question 1.4 Find the number of diagonals that can be drawn in a regular polygon having 37 sides.

End of Round 1 Please send your next cipherer forward at this time

Round 2 Question 1

Question 2.1 A locomotive is pulling 10 supply cars. 5 of the cars are carrying coal, 3 of the cars are carrying lumber, and the rest of the supply cars are carrying oil. How many distinct ways can you arrange all the cars behind the locomotive if there is also a caboose and 2 identical passenger cars that must be connected behind the last supply car? (note: the caboose does not necessarily have to be the last car on the train)

Round 2 Question 2

Question 2.2 Given the figure below, find xy. Assume all angles that look like right angles are right. x z y 8 8

Round 2 Question 3

Question 2.3 f(x) = 4x + 5 g(x) = x 2 – 2 Find f(g(3x + 6))

Round 2 Question 4

Question 2.4 Find the remainder when 4x 3 – x 4 + x 2 + 3x is divided by x-3.

End of Round 2 Please send your next cipherer forward at this time

Round 3 Question 1

Question 3.1 Find (log 6 9)(log 9 12)(log 12 15)…(log )

Round 3 Question 2

Question 3.2 How many consecutive zeros does have at the end?

Round 3 Question 3

Question 3.3 Write (1 + i) -4 in a + bi form.

Round 3 Question 4

Question 3.4 Find the units digit of

End of Round 3 Please send your next cipherer forward at this time

Round 4 Question 1

Question 4.1 Find the sum of the integral divisors of 144.

Round 4 Question 2

Question 4.2 Find the square of the length of the apothem of a regular hexagon with perimeter 60

Round 4 Question 3

Question 4.3 If A = find the determinant of A 2 – 4A

Round 4 Question 4

Question 4.4 The 3rd term of a geometric progression is 8/3. The 5th term is 1/6. Find the sum of the first 4 terms.

End of Ciphering

Extra Question #1

Question E.1 What is the probability that two integers from [1,10] selected randomly with replacement can be the numerator and denominator, respectively, of a proper fraction in lowest terms?

Extra Question # 2

Question E.2 7A = 2B 3B = 8C C = 15D 2D = 5E Find A/E