MATHCOUNTS
2008
• National Competition •
Countdown Round
Problems 1^
This section contains problems to be used in
the Countdown Round.
National Sponsors
Lockheed Martin * Raytheon Company
Northrop Grumman Foundation * Texas Instruments Incorporated
National Society of Professional Engineers * 3M Foundation
General Motors Foundation * CNA Foundation * ConocoPhillips
National Council of Teachers of Mathematics
LOCK nee o Mauri t
MATHCOl'NTS
National Com
tition
Sponsor
2006-2008
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation
Copyright MATHCOUNTS, Inc. 2008. All rights reserved.
Given the information shown in the
table, what percent of the human
body is made up of elements other
than oxygen, carbon and hydrogen?
Elements in the Human Body
Element
Percent
Oxygen
65%
Carbon
18%
Hydrogen
10%
1.
2. The values of a, b, c and dare I , 2, 3 and 4, but not necessarily in
that order. What is the largest possible value of the sum of the four
products ab, be, cc/and dal
2.
3. Alexia designed a logo 2 inches wide and 1.5 inches tall to be
used on her school's notebooks. The school wants the logo on the
notebook to be 8 inches wide. How tall, in inches, will it be if it is
enlarged proportionately?
3.
4. Tickets to a concert cost $16 for adults and $13 for children. A group
of adults and children spent $154 for its tickets. How many adults
were in the group?
4.
5. A square of area 6 square units is split into rectangles of
areas 3,2 and 1 square units, as shown. What is the ratio
of the length of the longer side of the gray rectangle to the
length of the shorter side of the gray rectangle? Express
your answer as a common fraction.
5.
6. What real number x satisfies the equation (3Jt)(4JC) = 7*?
6.
7. BoatWorks built 3 canoes in January of this year and then each
'? subsequent calendar month they built twice the number of canoes
they had built the previous month. How many total canoes were built
by BoatWorks by the end of March of this year?
7.
8. Bob can run 1 mile in the same time that Gill can run f of a mile.
Cindy can run \ of a mile in the same time that Gill can run 1 mile.
If each person runs at those rates, how many miles will Cindy run in
the time it takes Bob to complete an 8-mile run?
8.
9. Each of the four students on each of 16 MATHCOUNTS teams wins
a math book valued at $125. What is the total value, in dollars, of all
of the math books received by the students?
9.
10. For integers a and b, what is the smallest possible positive value of
8a+122»?
10.
11. The Norman H.S. math club has twice as many male members as
female members. Twenty percent of the male members and 30% of
the female members participated in a math contest. What fraction
of those members participating in the math contest were female?
Express your answer as a common fraction.
11.
12. The first three terms of an arithmetic sequence are 1,10 and 19,
respectively. What is the value of the 21st term?
12.
Copyright MATHCOUNTS, Inc. 2008. All rights reserved. 2008 National Countdown Round
13. Forty-nine of the first 50 positive integers are in a list whose sum is
1257. Which one of the first 50 positive integers is missing from the
list?
13.
14. What is the slope of a line perpendicular to the line whose equation is
j — y = 1 ? Express your answer as a common fraction.
14.
9x 13
15. What is the sum of all possible solutions of 7 T + ~ ~
250
16. When Julian spends his weekly allowance at the bookstore, his father
lets him buy a book whose regular price is double his allowance.
However, he gets a 10% discount on the price of the book. What
fraction of the final purchase price is Julian's weekly allowance?
Express your answer as a common fraction.
16.
1 7. Sandy is thinking of a two-digit perfect square, :c. The sum of the digits
of x is also a perfect square. What is the sum of the possible values
of*?
17.
1 8. Each term of a sequence, after the first term, is inversely proportional
to the term preceding it, and the constant of proportionality stays the
same. If the first term is 2 and the second term is 5, what is the
12th term?
1 8.
19. If 4 daps = 7 yaps and 5 yaps = 3 baps, how many daps equal
42 baps?
19.
20,^Math City has eight streets, all of which are straight. No street is
parallel to another street. One police officer is stationed at each
intersection. What is the greatest number of police officers needed?
20.
21. The equation * 2 -fo-12 = 0 has only integer solutions for certain
positive integers k. What is the sum of all such values of £?
21.
22. Paint is to be mixed so that the ratio of red paint to white paint is 3 to
2. If Ben wants to make 30 cans of the mixture and all cans hold the
same volume of paint, how many cans of red paint will he need?
22.
23. Kyle's birthday party in 2007 cost $d dollars per person and/? people
attended. In 2008, the cost per person doubled and the number of
people attending the party also doubled. What is the ratio of the total
cost of the party in 2007 to the total cost of the party in 2008? Express
your answer as a common fraction.
23.
24. An integer is randomly selected from the integers 0 through
99, inclusive. What is the probability that the integer is in the range
10 through 20, inclusive? Express your answer as a common fraction.
24.
25. The two solutions of the equation x2 + bx + 1 8 = 0 are in the ratio of
2 to 1 for some values of b. What is the largest possible value of bl
25.
Copyright MATHCOUNTS, Inc. 2008. All rights reserved. 2008 National Countdown Round