Rules of Engagement Please turn off all cell phones while Math Bowl is in progress. The students participating in Rounds 1 & 2 will act as checkers for one another, as will the students participating in Rounds 3 & 4. There is to be no talking among team members once the round has begun. Any pairs caught talking, even between questions, will be ejected from the competition.

Checkers are more than welcome to take a chance that the answer their teammate gave is also correct, though it doesn’t appear as a possible answer. However, keep in mind that if the answer is in an unacceptable form or otherwise incorrect, points will be deducted from the team score according to how many points would have been received if the answer was correct. (5 points will be deducted for an incorrect first place answer.)

Checkers, please remember that multiplication and addition are commutative. Correct solutions not placed in the given answer space are not correct answers! Rationalize all denominators. Reduce all fractions, unless the question says otherwise. Do not leave fractions as complex fractions. Use only log base 10 or natural log.

It is only necessary to write an equation when asked for an equation or a function. Answers of the form are acceptable, unless both answers are rational. Use interval notation for domains and/or ranges. When units are given in the problem, units are required in the answer. Good luck, and most importantly, have fun!

2005 Math Bowl Varsity Round 1

Practice Problem – 15 seconds Let. Find.

Problem 1.1 – 25 seconds Find the ordered triple that satisfies the system

Problem 1.2 – 25 seconds Several logs are stored in a pile with 20 logs on the bottom layer, 19 on the second layer, 18 on the third, and so on. If the top layer has one log, how many logs are in the pile?

Problem 1.3 – 30 seconds Let and. Find the polynomial.

Problem 1.4 – 20 seconds For the sets, and, find..

Problem 1.5 – 20 seconds If the point is on the graph of, find a.

Problem 1.6 – 15 seconds Write as a simple trigonometric function.

Problem 1.7 – 25 seconds Determine the domain of the function

Problem 1.8 – 15 seconds Find the length of x.

Problem 1.9 – 30 seconds Find the area of the parallelogram in the plane with vertices

Problem 1.10 – 25 seconds Solve for y :

Problem 1.11 – 30 seconds Find the arc length corresponding to a central angle of on a circle with radius 7 cm.

Problem 1.12 – 20 seconds Calculate

Round 2

Practice Problem – 25 seconds Simplify

Problem 2.1 – 15 seconds Simplify

Problem 2.2 – 20 seconds Simplify completely.

Problem 2.3 – 25 seconds Let. Find.

Problem 2.4 – 15 seconds Find the exact value of.

Problem 2.5 – 15 seconds What are the next two terms in the sequence A, c, E, g, …

Problem 2.6 – 35 seconds Find the center of the ellipse

Problem 2.7 – 20 seconds Find the roots of

Problem 2.8 – 25 seconds If,, and, find.

Problem 2.9 – 20 seconds Find the next term of the sequence 20, 17, 13, 8, …

Problem 2.10 – 15 seconds According to the rational root theorem, what are the possible rational roots of

Problem 2.11 – 25 seconds If, find.

Problem 2.12 – 35 seconds For what interval(s) of x does produce real y values?

Round 3

Problem 3.1 – 30 seconds The area of an equilateral triangle varies directly with the square of the length of a side. Find the constant of proportionality.

Problem 3.2 – 30 seconds Solve in the interval.

Problem 3.3 – 20 seconds Calculate

Problem 3.4 – 25 seconds Find the length of CD in terms of x.

Problem 3.5 – 20 seconds Evaluate

Problem 3.6 – 30 seconds Find the inverse of

Problem 3.7 – 20 seconds Find the polar equation for the Cartesian equation

Problem 3.8 – 30 seconds Evaluate on the interval.

Problem 3.9 – 40 seconds Let and. Find.

Problem 3.10 – 30 seconds Find the coefficient of in the expansion of.

Problem 3.11 – 25 seconds How many times can the face 5 be expected to occur in a sequence of 2016 throws of a fair die?

Problem 3.12 – 25 seconds If, and, find.

Round 4

Problem 4.1 – 20 seconds Find.

Problem 4.2 – 35 seconds Expand into partial fractions.

Problem 4.3 – 20 seconds Let. Find, with only positive exponents in the answer.

Problem 4.4 – 25 seconds Find the sum of the first five multiples of 4.

Problem 4.5 – 20 seconds A couple is planning their wedding. They can select from 2 different chapels, 4 soloists, 3 organists, and 2 ministers. How many different wedding arrangements are possible?

Problem 4.6 – 25 seconds Find the distance between the points and.

Problem 4.7 – 15 seconds If and, find.

Problem 4.8 – 35 seconds Find

Problem 4.9 – 35 seconds Find c in the interval such that if.

Problem 4.10 – 30 seconds Evaluate

Problem 4.11 – 20 seconds Find the slope of the tangent line to the graph of, at the point.

Problem 4.12 – 45 seconds A gum manufacturer randomly puts a coupon in 1 of every 5 packages. What is the probability of getting at least one coupon if 4 packages are purchased?