Here are brief descriptions of
the various events, with a few sample problems. The Math Bowl and the Test
of Ingenuity are the more serious events; the winners of these receive
the bigger prizes. The Relay Race and TurboBuzz are more for fun, but there
will be prizes for the winners in these events as well.
The Math Bowl is divided into Open rounds and Invitational rounds. During the Open rounds, teams compete in pairs. A moderator asks a question to a single team at a time, alternating teams. The team has a fixed time limit (approximately 45 seconds, sometimes more) to respond. Each correct response gets a point.
The teams with the highest total of Open round points move on to the Invitational rounds. This is an elimination tournament where the teams again compete in pairs, but the same question is asked of both teams. The first team to activate a buzzer gets to respond. If the answer is correct, the team gets a point and the moderator asks a new question to both teams. If the answer is wrong, the team loses a point and the other team gets a chance to answer, etc., until the time runs out. At the end of each round, the team with the greater number of points moves on to the next round and the other team is eliminated. The questions get harder each round, and eventually one team is left, the winner of the tournament.
Some Easy Bowl Questions from past years
1. What is the smallest positive integer, each digit of which is a 2, which is divisible by 9?
2. Suppose a rhombus has sides of length 1 and area 1/2. Find the angle formed by the two diagonals of the rhombus.
3. Suppose that A-B=0, B-C=0,
C-D=0, ..., Y-Z=0, and A+Z=26. Find M.
Some Medium Bowl Questions from past years
1. At a certain chess tournament, there are 10 people, and each person plays exactly one game with each other person. How many games are played?
2. Find integers m and n such that 1/m+1/n = 1/7.
3. If we graph y = 
then what is the name of the curve we get?
4. The median of a set of 11
real numbers is 0. What is the maximum number of positive integers in this
set?
Some Hard Bowl Questions from past years
1. Express the following as a fraction in simplest form:
(cos 20°) (cos 40°) (cos 80°).
2. Express the infinite repeating base-two decimal 0.01010101... as a (base-ten) fraction in lowest terms.
3. In chess, the rook attacks any piece located on the same row or column. If two rooks are placed randomly on two different squares of an 8 x 8 chessboard, what is the probability that they are attacking one another?
4. In polar coordinates, if we graph r = sinø, then what is the name of the curve we get?
5. An icosahedron is a polyhedron built out of 20 equilateral triangles which meet five to a vertex. How many vertices does the icosahedron have?
6. Find a three-digit positive integer that is divisible by 7 when you subtract 7 from it, is divisible by 8 when you subtract 8 from it, and is divisible by 9 when you subtract 9 from it.
7. Find the ordered pair (x,y) that satisfies the system
6751x + 3249y = 26751,
3249x + 6751y = 23249.
If you like those, here are more Bowl questions.
This is a multiple-choice, 50-minute
exam. Many of the questions will stress ingenuity, rather than just memorization
or complicated calculation. The easier questions will be comparable to
AHSME (American High School Math Exam) questions in difficulty, while the
harder questions will be at least as hard as AIME (American Invitational
Math Exam) questions. The exam is designed to have enough fun, interesting
questions for everyone, yet also a few super-hard problems to challenge
the very best students.
Sample Questions from the 1994 Exam
1. For non-zero real numbers, let f(x) = 1/x. Compute f(f(f(…(f(2))…))), where the above expression contains 20 fs.
2. Mary and Jane start at the same point, and run around a track in the same direction. It takes Mary 2 minutes to complete each lap, and Jane 2 minutes and 15 seconds to complete a lap. After 1 hour, how many times has Mary passed Jane (not including when they had just started running)?
3. In a circle of radius 1, a
chord of length 
is drawn,
dividing the interior of the circle into two regions. Find the area of
the smaller region.
4. Assume that the Earth is a perfect sphere with radius 4,000 miles. Suppose you start at latitude 30 degrees north of the equator (in other words, if your position is P, the center of the earth is O, and the North Pole is N, then the measure of angle PON is 60 degrees). If you travel east until you return to your original position, then how many miles will you travel?
5. If a+b+c=0 and a3+b3+c3=27, find abc.
6. For a deck containing an even number of cards, define a “perfect shuffle” as follows: divide the deck into two equal halves, the top half and the bottom half, then interweave the cards one-by-one between the two halves starting with the top card of the bottom half, then the top card of the top half, etc. For example, if the deck has 6 cards, labeled “123456” from top to bottom, after a perfect shuffle the order of the cards will be “415263”. Determine the minimum (positive) number of perfect shuffles needed to restore a 94-card deck to its original order.
exams with answers.
This is a relaxed, informal but fast-paced event where 3 to 6 students work together to solve as many problems as possible in a short amount of time. The best teams figure out a strategy to divide up resources in a cooperative way. The teams that let their strongest member do all the work invariably have difficulty!
A Sample Team Contest
1. Simplify (1+i)97/(1-i)94,
where i= 
. Your answer should
be in the form a+bi, where a and b are real numbers.
2. Find all solutions to the equation ||x-3| - |x-13|| =2.
3. Four people, A, B, C and D, are on one side of a river and must cross an old bridge at night to get to the other side. The bridge is too weak to carry more than two people at a time. It is too dark to cross the bridge without a flashlight, and the four people have only one flashlight. Because of this, whenever two people cross the bridge, they must walk at the pace of the slowest person, for the flashlight is weak and cannot illuminate very far. A, B, C and D can cross the bridge in 5, 10, 20 and 25 minutes, respectively. How can all four cross the bridge in one hour? No tricks allowed, like carrying someone or throwing the flashlight, etc. It can be done legitimately.
4. Let S be the set of positive integers with the largest possible product which add up to 1996. What is the product?
5. A census-taker knocks on a door, and asks the man inside how many children they have and how old they are.
"I have three daughters, their ages are integers, and the product of the ages is 36," says the father.
"That's not enough information," responds the census-taker.
"I'd tell you the sum of their ages, but you'd still be stumped."
"I wish you'd tell me something more."
"Okay, my oldest daughter Annie is mathematically inclined."
What are the ages of the three daughters?
Traditional "Buzz" is a game sometimes played in elementary school. The students form a circle, calling out the numbers in order, “One, two, three, ...”, following the rule that if the number contains a 7 in it or is a multiple of 7, one says” buzz” instead of the number. For example, the first student says “one,” the next says “two,” ..., the sixth says “six” and the seventh one says “buzz”, etc. If a student says the wrong thing, then he or she leaves the circle. The game continues until one person is left, the winner.
TurboBuzz begins like regular Buzz, but from time to time the moderator
introduces new rules which have higher priority. For example, if after
“15” is called out, the moderator announces, “Say 'meow' for primes,” the
next student will say “16” but the one after that must say “meow”, since
“meow” takes priority over the original “buzz” rule. Once a rule is added,
it stays valid, unless two rules clash, in which case the most recent rule
takes priority. As with regular Buzz, the winner of a TurboBuzz game is
the last person standing.
Last Modified 4 March, 2003.
These pages maintained by Christine Liu .
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