MATHCOUNTS
2008 Gauss1181
AoPS Competition #2
Sprint Round
Problems 1-30
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
Name_________________________________________________________________________
Team_________________________________________________________________________
This round of the competition consists of 30 problems. You will
have 40 minutes to complete the problems. You are not allowed to
use calculators, books or any other aids during this round. If you
are wearing a calculator wrist watch, please give it to your proctor
now. Calculations may be done on scratch paper. All answers
must be complete, legible and simplified to lowest terms. Record
only final answers in the blanks in the right-hand column of the
competition booklet. If you complete the problems before time is
called, use the remaining time to check your answers.
In each written round of the competition, the required unit for the
answer is included in the answer blank. The plural form of the unit
is always used, even if the answer appears to require the singular
form of the unit. The unit provided in the answer blank is the only
form of the answer that will be accepted.
Total Correct
Scorer’s Initials
1. During recess time in elementary school, Darryl wants to do one of four indoor activities
(Lincoln Logs, computer on AoPS, chess, or Scrabble), one of three outdoor activities
(basketball, soccer, or jump rope), and two of five pranks (drawing on the blackboard,
stealing a potato chip from gauss1181, messing up Mark’s girlfriend’s hair, trying to
guess his friend’s password, or crack the secret code on the teacher’s grade sheet). In how
many ways can he combine his activities?______________________________________
2. In a survey of 100 students, 23 played football, 20 played basketball, and 17 played
soccer. 12 students played football and basketball, 6 students played basketball and
soccer, and 5 students played football and soccer. 3 students played all 3 sports. How
many students did not play football, basketball, or soccer?_________________________
3. Weina and her friend, Isabella, were once the same height. Since a year, Isabella grew
10% taller while Weina grew half as many inches as Isabella. Isabella is now 66 inches
tall. How tall, in inches, is Weina now?________________________________________
4. Wenyu and Mario are going to play a game of foosball. The first player to score 4 goals
wins. Wenyu is better, and has a 70% chance of scoring each goal. What is the
probability that Mario wins?_________________________________________________
5. In an AoPS FTW Countdown tourney, 16 players battle each other in a single-elimination
bracket. A player is eliminated if he/she loses a game. Among the players, Bobby is the
best player, Kevin is the second best, and Isabella is third best. The better player of each
match always wins. If the players were all placed in the bracket at random, what is the
probability that Isabella is the runner-up of the tournament? (she gets to the finals, but
loses) Express your answer as a common fraction._______________________________
6. Farmer Shen and Farmer Tang agree that cows are worth $200 and chickens are worth
$90. If one farmer owes the other money, he pays the debt in cows or chickens (so a $110
debt can be paid with one cow, with one chicken received in change). What is the
smallest possible debt that can be resolved in this way?___________________________
7. Runpeng runs 30% faster than his brother, Calvin. Calvin runs at a rate of 5 miles per
hour. In a 100 mile marathon, how much longer would it take Calvin to finish the course
than Runpeng?___________________________________________________________
8. A triangle has sides of 10, 10, and 12. What is the area of the triangle?_______________
9. Robert has a bag of gold and silver coins. He has 7/5 as many gold coins as silver coins.
If he has between 80 and 90 coins, how many silver coins does he have?______________
10. In a certain classroom, ¾ of the students have blonde hair. This is equivalent to the
number of students who like math. 8 students don’t like math. What is the largest number
of students possible who both like math and have blonde hair?______________________
11. Alan, Hao, Yongyi, and Xiaoyu share their money together. Alan has $12 more than Hao.
Yongyi has $6 more than Xiaoyu. Xiaoyu has $5 more than Alan. If the boys divide up
their money so that each person gets the same amount of money, how much money does
Yongyi have to give away?__________________________________________________
12. The sum of two numbers is 1337. Their product is 442,200. Find the smaller of the two
numbers.________________________________________________________________
13. How many three-digit perfect squares are divisible by 11?_________________________
14. The 2008 AMC 12 is scored by awarding 6 points for each correct answer, 1.5 points for
each question left blank, and 0 points for an incorrect answer. After looking over the
test’s 25 problems, Dhaivat decided to attempt the first 21 problems and leave the last 4
unanswered. If he wants to get a score of 100 or more, what is the greatest number of
problems he could answer incorrectly?_________________________________________
15. Andrew’s mathematics test has 50 problems. It has 10 algebra problems, 12 geometry
problems, 15 number theory problems, and 13 probability problems. Andrew answered
80% of the algebra questions right, 75% of the geometry problems correctly, 60% of the
number theory questions right, and 4 of the probability problems right. What was his
score on the overall test?____________________________________________________
16. An abundant number is a number whose proper divisors have a sum greater than the
number. For example, 12 is an abundant number because 1+2+3+4+6=16>12. What is
the largest two-digit abundant number?________________________________________
17. Uthsav has 48 square feet of flooring that he needs to build a rectangular fence around.
He wants to minimize the amount of fencing he uses. What is the minimum perimeter of
fencing a 48 square foot floor could be enclosed in?______________________________
18. One day, Erik caught, marked, and released 191 fish from a lake. The next day, Erik
observed the lake again and saw 84 fish, where 12 of them had been marked the previous
day. What is the best estimate for the total number of fish in the lake?________________
19. Hexagon ABCDEF has apothem 6. What is its area? Express your answer as a common
fraction (if necessary) and in simplest radical form._______________________________
20. Mark’s basketball team has 10 members. In how many ways can they choose a starting
lineup of 2 guards, 2 forwards, and a center, if Mark must be a forward?______________
21. Billy Bob has $30 he saved from his allowance. If he continues to double his allowance
after each week, after how many weeks would he have more than $1337?_____________
22. Jorian wants to form 4-digit integers using only the digits 1, 2, 3, and 4. He is allowed to
repeat digits. What is the probability that he forms a 4-digit integer where at least one
digit repeats? Express your answer as a common fraction._________________________
23. Angie wanted to use her class as a sample for predicting the number of kids who like
math. In her class of 24 students, she counted 20 people who like math. Her school has
360 kids in all. How many students would Angie predict in her school who like
math?___________________________________________________________________
24. Eduardo has 10 friends. In how many ways can he pick 3 of them to sit with him at lunch,
if 7 of his friends are boys and 3 of them are girls, and he cannot pick a combination of
all boys or all girls?________________________________________________________
25. Triangle ABC has AB=5, BC=4, and angle ABC measures 60 degrees. What is the length
of AC? Express your answer in simplest radical form.____________________________
26. What is the units digit of 20082008  20072007 ?___________________________________
27. If 6 mammy traps=2 pappy traps and 1 Yonker is equal to 50 pappy traps, how many
mammy traps are in a Yonker?_______________________________________________
28. Alex, Evan, and Shaunak are playing a coin game. Alex goes first, followed by Evan,
then followed by Shaunak. The first person to flip heads wins. What is the probability
that Alex wins? Express your answer as a common fraction.________________________
29. How many degrees are in one interior angle of a 20-sided regular polygon?___________
30. Ten cards from a standard deck are being chosen, numbered 1 through 10, and placed in a
hat. Two are being drawn. What’s the probability that they’re both face cards from the
same suit? Express your answer as a common fraction.____________________________
MATHCOUNTS
2008 Gauss1181
AoPS Competition #2
Target Round
Problems 1-8
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
Name________________________________________________________________________
Team________________________________________________________________________
This round of the competition consists of 8 problems, which will be presented in pairs. Work on
one pair of problems and answers will be collected before the next pair is distributed. Record
only final answers in the blank spaces of the competition booklet. You will have 6 minutes per
pair of problems. If you finish early, use the remaining time to check your answers. This round
assumes the use of calculators, and calculators may also be done on scratch paper, but no other
aids are allowed.
Total Correct
Scorer’s Initials
1. Carl Friedrich Gauss, gauss202, and gauss1181 are siblings. Carl Friedrich Gauss is 2
years older than gauss1181, who is 1/3 of gauss202’s age. The sum of the three siblings’
ages is 78. What is the product of their ages?_________________________
2. Mr. Dunbar planted 3 trees 2 years ago, 10 trees 5 years ago, and 15 trees 10 years ago.
Twenty years later, he plants 6 more trees. At that time, what is the mean age of his trees?
Express your answer to the nearest whole number.___________________
3. Junu, Anderson, and Bob, whose last names are (not necessarily in respective order)
Wang, Smith, and Bae, have favorite colors red, blue, and brown. Each person’s favorite
color has the same number of letters as either his first or last name. No one has the same
number of letters in his first and last name. What is the name of the person whose favorite
color is red?________________________________________
4. In triangle ABC, AB=6 and BC=7. In triangle ADE, BD=3 and CE=5. What is the ratio
of the area of triangle ABC to the area of triangle ADE? Express your answer as a
common fraction._____________________________________________________
5. Nguyen and Jingfei are running around a circular track. Nguyen runs at 5 miles per hour,
and Jingfei runs at 4 miles per hour. They both start from the same point on the track and
run in opposite directions. If the two first are at opposite ends of the track after 5 hours,
how long is the circular path of the track?________________________
6. Circles A, B, and C are externally tangent. Circles A and B each have radius 2 while
circle C has radius 1. What is the radius of the circle that circumscribes triangle ABC?
Express your answer as a common fraction in simplest radical form._________
7. A 60-liter salt water solution consists of 10% salt and the rest pure water. How much
pure water must be added to the mixture so that it consists of 8% salt?_____________
8. Gregory has 4 blue marbles, 5 green marbles, and 6 red marbles. He plans to put x more
blue marbles and y more green marbles in his possession so that the probability of
picking a red marble is 1/5. However, he wants to make sure that there is an equal chance
of drawing a blue and green marble. How many green marbles does Gregory need to add
to his inventory?_____________________________________________
MATHCOUNTS
2008 Gauss1181
AoPS Competition #2
Team Round
Problems 1-10
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
Team_________________________________________________________________________
____________________________________, Captain
___________________________________________
_____________________________________________
_____________________________________________
This section of the competition consists of 10 problems which the team
has 20 minutes to complete. Team members may work together in any
way to solve the problems. Team members may talk during this section
of the competition. This round assumes the use of calculators, and
calculations may also be done on scratch paper, but no other aids are
allowed. All answers must be complete, legible and simplified to lowest
terms. The team captain must record the team's official answers on his/her
own problem sheet, which is the only sheet that will be scored.
Total Correct
Scorer’s Initials
1. Twenty-two people, each weighing an average of 150 pounds, want to climb an elevator
that can support up to 1 ton of the people’s weight at a time. The first elevator arrives,
and the maximum possible number of people boards the elevator. How many people have
to wait for the second elevator?______________________________________________
2. Albert, Betsy, and Carl are mowing a rectangular lawn. Albert and Betsy can mow the
lawn together in 4 hours, Betsy and Carl can do it in 3 hours, and Albert and Carl can do
it in 5 hours. If they all start mowing the lawn together, how long would it take for them
to finish the job if Carl gets tired after 1 hour and refuses to continue?________________
3. Superhero Jason Wang enjoys his job involving catching Internet impersonators. Over a 7
day period, one person commits an impersonating crime each day. Jason checks for
impersonators at the end of every day, and has a 4/5 chance of catching every
impersonator that committed a crime so far. What is the probability that, at the end of 7
days, all 7 impersonators would be caught?_____________________________________
4. Chen, Shen, Tang, Yang, Li, and Nguyen all stay for one night at the Mathletes Hotel and
Resort. They have reserved 4 rooms, all of which are distinguishable. Each room can
hold up to 2 people, and Shen has to have a room of his own. In how many ways can the
six friends reserve their rooms?______________________________________________
5. What is the remainder when 20091337 is divided by 5?_____________________________
6. The AM-GM Company decides to invest $5000 in its shares compounded continuously.
With 6% interest, how much would the company have invested after 4 years?__________
7. Brian is rowing his boat when he notices a leak starting to come into the boat. The leak
causes water to rush into the boat at a rate of 5 liters per minute. While he rows, Brian
tries to bail water out of the boat as fast as he could, before it sinks. If he rows at a rate of
4 miles per hour, what’s the slowest speed, in liters per minute, he can bail water out of
the boat before it sinks before Brian can cross the 10,000 mile river he’s trying to row
across?________________________________________________________________
8. In triangle ABC, B is a right angle and the ratio of BC to AB is 4:3. The area of ABC’s
circumscribed circle is 100pi square inches. What is the area of triangle ABC__________
9. How many perfect numbers less than 1000 exist?________________________________
10. Square ABCD has side length 10. Additional squares are formed by using midpoints of
square sides as smaller squares’ vertices. If 10 squares are being drawn in total, what’s
the area of the smallest square?______________________________________________