DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

Chapter Competition 2004 Sprint Round Problems 1–30

MATHCOUNTS

Name School

DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

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.

Total Correct Scorer’s Initials

Founding Sponsors National Sponsors

CNA Foundation The Dow Chemical Company Foundation National Society of Professional Engineers General Motors Foundation

National Council of Teachers of Mathematics Lockheed Martin

National Aeronautics and Space Administration NEC Foundation of America

Texas Instruments Incorporated 3M Foundation

1. There are three 7th-grade math classes, each with 24 students.

No student is taking more than one math class. How many students are in 7th-grade math classes altogether?

2. A particular triangle has sides of length 14 cm, 8 cm and 9 cm.

In centimeters, what is the perimeter of the triangle?

3. The sum of the squares of two positive integers is 73. What is the sum of the two positive integers?

4. Kaleb defined a clever integer as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 9.

What fraction of all clever integers is divisible by 27? Express your answer as a common fraction.

5. Based on this graph, what percent of viewers watched one hour or more of television?

6. Solve for x: ¹ 2

1 3

− =1 x .

7. The points (x, y) represented in this table lie on a straight line.

The point (28, t) lies on the same line. What is the value of t ?

1. ________________

2. ________________

3. ________________

4. ________________

5. ________________

6. ________________

7. ________________

x y 1 7 3 13

8. Place a digit 0 through 9 in each blank so the equation below is true. What is the sum of the six digits you placed in the blanks?

( _6_ ,3_1 ) - ( 76,_35 ) = ( _9,28_ )

9. A pizza costing $17.60 is cut into 16 equal pieces. Brad eats six pieces. In dollars, how much was the portion of the pizza that Brad ate? Express your answer as a decimal to the nearest hundredth.

10. This net is folded into a cube. When the cube is rolled, the lateral product is the product of the numbers on the four lateral faces. The numbers on the top and bottom faces are not included in the product. What is the greatest possible lateral product for this cube?

11. Rick has a cup containing at least one penny, at least one nickel and at least one dime. The total value of the coins in the cup is

$1.25. What is the sum of the largest possible number of nickels in the cup and the smallest possible number of nickels in the cup?

12. On Friday, a snowboard originally priced at $100 was

discounted 50%. On Monday, that sale price was reduced by 30%. In dollars, what is the price of the snowboard after the Monday reduction?

13. The sum of two numbers x and y is 153, and the value of the fraction x

y is 0.7. What is the value of y - x?

8. ________________

9. ________________

10. ________________

11. ________________

12. ________________

13. ________________

14. Given the set {3, 6, 12}, how many unique values can be created by forming fractions ^xy such that x and y are distinct members of the given set?

15. There are eight furlongs in a mile. There are two weeks in a fortnight. The British cavalry traveled 2800 furlongs in a fortnight. How many miles per day did the cavalry average?

16. The number 121 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 100 and 500?

17. How many diagonals does a regular seven-sided polygon contain?

18. A detective is standing at a corner D. A witness tells the detective,

“The shortest distance from you to the suspect is exactly four blocks.”

How many different intersection points on this map are exactly four blocks from D, when traveling the shortest path along the segments of the grid?

19. Define ^{a b}∆ =^a2−^b. What is the value of 2

3 8 3 8

14. ________________

15. ________________

16. ________________

17. ________________

18. ________________

19. ________________

20. Five consecutive terms of the Warloe sequence are given below.

In the Warloe sequence, each term after the third term is the sum of the previous three terms. The 8^th term has been identified.

What is the sum of the 1^st term and the 13^th term in the Warloe sequence?

... , 1, 2, 3, 6, 11, ...

21. If − + <2a 1 13, what is the sum of the distinct possible integer- values of a?

22. Ilian will randomly choose a value for p from the integers 0 through 13, inclusive, and will make ₁₃^p the coordinate of

point A. The coordinate of point B is ²₃. Points A and B will be on the same number line. What is the probability that point A and point B will be less than ¹₅ of a unit from each other?

Express your answer as a common fraction.

23. This graph depicts Speed versus Time for a car trip. Which scenario below, (A, B, C

or D), best describes this graph?

A. A car was going up a hill, then along a level street, and then down the hill.

B. A car was going forward, then stopped, and then drove backwards.

C. A car was speeding up, then advanced at a constant speed, and then decreased in speed.

D. A car was heading northeast, then went east, and then went southeast.

24. How many terms of the arithmetic sequence 88, 85, 82, ...

appear before the number -17 appears?

20. ________________

21. ________________

22. ________________

23. ________________

24. ________________

8^th term

25. The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12?

26. Starting at the M in the center, you may move left, right, up or down to an adjoining letter. How many distinct paths can be followed to spell the word MATH?

27. A regular pentagon and a regular hexagon are coplanar and share a common side

AD, as shown. What is the degree measure of angle BAC?

28. The coordinates of the vertices of a parallelogram are (10, 1), (7, -2), (4, 1) and (x, y). What is the sum of the distinct possible values for x ?

29. Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy’s list?

30. A particular right square-based pyramid has a volume of 63,960 cubic meters and a height of 30 meters. What

is the number of meters in the length of the lateral height (AB) of the pyramid? Express your answer

to the nearest whole number.

25. ________________

26. ________________

27. ________________

28. ________________

29. ________________

30. ________________

Forms of Answers

The following list explains acceptable forms for answers. Coaches should ensure that Mathletes^® are familiar with these rules prior to participating at any level of competition. Judges will score competition answers in compliance with these rules for forms of answers.

All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction in the form , where a and b are natural numbers, and GCF(a,b) = 1. In some cases the term “common fraction” is to be considered a fraction in the form , where A and B are algebraic expressions, and A and B do not share a common factor. A simplified “mixed number” (“mixed numeral,” “mixed fraction”) is to be considered a fraction in the form N , where N, a and b are natural numbers, a < b and GCF(a,b) = 1.

Examples:

Problem: Express 8 divided by 12 as a common fraction. Answer: ²₃ Unacceptable: ⁴₆, ₁₂⁸ Problem: Express 12 divided by 8 as a common fraction. Answer: ₂³ Unacceptable: ⁶₄, ¹²₈ Problem: Express the sum of the lengths of the radius and the circumference of a circle with a diameter of as a common fraction in terms of p. Answer:

Problem: Express 20 divided by 12 as a mixed number. Answer: 1 Unacceptable: 1 ,

Ratios should be expressed as simplified common fractions, unless otherwise specified. Examples:

Simplified, Acceptable Forms: , , Unacceptable: 3 , , 3.5, 2:1

Radicals must be simplified. A simplified radical must satisfy the following conditions: 1) no radicands have perfect square factors other than one; 2) no radicands contain fractions; and 3) no radicals appear in the denominator of a fraction. Numbers with fractional exponents are not in radical form. Examples:

Problem: Evaluate × . Answer: 5 Unacceptable:

Answers to problems asking for a response in the form of a dollar amount or an unspecified monetary unit (e.g. “How many dollars...,” “What is the amount of interest...”) should be expressed in the form ($) a.bc, where a is an integer and b and c are digits. The only exceptions to this rule are when a is zero, in which case it may be omitted, or when b and c are both zero, in which case they may both be omitted. Examples:

Acceptable: 2.35, 0.38, .38, 5.00, 5 Unacceptable: 4.9, 8.0

Units of measurement are not required in answers, but must be correct if given. When a problem asks for an answer expressed in a specific unit of measure, equivalent answers expressed in other units are not acceptable.

For example, if a problem asks for the number of ounces and 36 oz is the correct answer, 2 lbs 4 oz will not be accepted. Similarly, if a problem asks for the number of cents and 25 cents is the correct answer, $0.25 will not be accepted.

Do not make approximations for numbers (e.g., p, , ) in the data given or in solutions, unless the problem says to do so.

Do not do any intermediate rounding (other than the “rounding” a calculator performs) when calculating solutions. All rounding should be done at the end of the calculation process.

Scientific notation should be expressed in the form a×10ⁿ where a is a decimal, 1 < |a| < 10 and n is an integer.

For example:

Problem: Write 6895 in scientific notation. Answer: 6.895×10³

Problem: Write 40,000 in scientific notation. Answer: 4×10⁴ or 4.0×10⁴

An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not be accepted. Whole number answers should be expressed in their whole number form.