m3 2008 diamond

Diamond Math Team Tryout

Category 1 Mystery Meet #3, January 2008 1. It�s your first day on the job as a bank teller. To test your wits, the management has a man come to your window to cash a check for $63. The man says, �Give me that amount in six bills, please, but no one-dollar bills.� List the values of the six bills you must give him. (You can omit the dollar signs, but please separate the numbers with commas.) Hint: You will need to use some bills that we don�t see very often. 2. How many rectangles are there in the grid below? Hint: There are more than 12. 3. Let A = 1, B = 2, C = 3, ..., Z = 26. The product value of a word is the product of the values of its letters. Find an English word that has a product value of 437.

Answers 1. _______________ 2. _______________ 3. _______________

Answers 1. _______________ 2. _______________ 3. _______________


Category 2 Geometry Meet #3, January 2008 1. The sum of the number of diagonals and the number of sides on a particular polygon is 28. How many sides are there on this polygon? 2. The area of square ABCD is 251 square units, the area of square DEFG is 690 square units, and the area of square AGHI is 934 square units. Which of the following conclusions is valid:

A. The measure of angle ADG is less than 90 degrees.

B. The measure of angle ADG is equal to 90 degrees.

C. The measure of angle ADG is greater than 90 degrees.

D. Pythagorus was a nice guy. 3. In the figure at right, angles ABC, CAD, and DAE are right angles. The measure of sides AB, AC, AD, and AE are 3 units, 5 units, 12 units, and 16 units respectively. How many units are in the perimeter of figure ABCDE? Give your answer to the nearest whole number. Hint: You will need to use the Pythagorean Theorem several times.

Answers 1. _______________ 2. _______________ 3. _______________

A B

C

D

E

C

B

I H

F

E

A G

D


Category 3 Number Theory Meet #3, January 2008 1. Evaluate the following expression. Note the different bases.

321Base Six + 321Base Five + 321Base Four = _______Base Ten 2. Maya�s calculator was stuck in scientific notation mode. When she finished computing the answer to a problem, the display looked like this: Maya thought that this means 9.53, but her friend Natalie told her (correctly) that it means 9.5 ×103. How far off is Maya�s answer? Express your answer as a regular number, not in scientific notation. Use a negative sign if Maya�s answer is too low and a positive sign if her answer is too high. 3. What base-ten fraction is equivalent to the base-three �decimal� 0.2121?

Answers 1. _______________ 2. _______________ 3. _______________

9.5 03


Category 4 Arithmetic Meet #3, January 2008 1. How many positive integers are less than 20084 ?

2. Evaluate the following expression. Write your answer as a common fraction in lowest terms.

213

−2

⋅ 113

3

⋅ 24

7

−1

627

0

3. How many different ways can the number 108 can be written as a sum of positive squares and positive cubes if there are no repeats and the number 1 is not used at all?

Answers 1. _______________ 2. _______________ 3. _______________


Category 5 Algebra Meet #3, January 2008 1. What inequality is represented by the graph below? Use the variable x on the left, the appropriate inequality sign in the middle, and a decimal on the right. 2. Find the sum of the two solutions that satisfy the equation below. Express your answer as a mixed number in lowest terms.

4x − 5 + 8 = 42 3. How many integer solutions are there to the double inequality below?

−8 < 7x +1323

≤ 11

Answers 1. _______________ 2. _______________ 3. _______________

0 1 2�1 �2


Solutions to Meet #3 Tryout, January 2008. Mystery 1. You need to use $2 bills to make this work. $50 + $5 + $2 + $2 + $2 + $2 = $63. 2. Counting systematically, you should find 12 rectangles that use just one region, 17 that use two regions, 10 that use three, 9 that use four, 7 that use six, 2 that use eight, 2 that use nine, and 1 rectangle that uses all twelve regions. That�s 60 rectangles in all. Alternatively, we can determine that there are 5C2 = 10 ways to choose two vertical lines and 4C2 = 6 ways to choose two horizontal lines. That�s 10 × 6 = 60 ways to make a rectangle. 3. The prime factorization of 437 is 19 × 23, which corresponds to the letters S and W. We will need an A (value 1) to make either the word SAW or WAS. Geometry 1. The chart at right shows the numbers of sides and diagonals on various polygons. The octagon, with 8 sides, has the correct sum of 28 sides and diagonals. 2. The Pythagorean Theorem is true for right triangles only. When the sum of the squares on the shorter sides is greater than the square on the longest side (as it is in this case), then the angle between those sides is less than 90 degrees, which is option A. 3. Each of the right triangles in the picture has whole-number side lengths. ABC is a 3-4-5 triangle, ACD is a 5-12-13 triangle, and ADE is a 12-16-20 triangle. The desired perimeter is 3 + 4 + 13 + 20 + 16 = 56 units. Number Theory 1. Converting each number to base ten, we get

321Base Six + 321Base Five + 321Base Four

= 3 × 36 + 2 ×6 +1( )+ 3 ×25 + 2 ×5 +1( )+ 3 ×16 + 2 ×4 +1( )= 108 +12 +1( )+ 75 +10 +1( )+ 48 + 8 +1( )=121+ 86 + 57 = 264Base Ten

2. The value of 9.53 is 9.5 × 9.5 × 9.5 = 857.375. The value of 9.5 × 103 is 9.5 × 10 × 10 × 10 = 9,500. Maya’s estimate is 9500 – 857.375 = 8642.625 too low, so the answer is –8642.625.

Polygon Sides Diagonals Sum Triangle 3 0 3 Square 4 2 6

Pentagon 5 5 10 Hexagon 6 9 15 Heptagon 7 14 21 Octagon 8 20 28 Nonagon 9 27 36


3. In a base-three �decimal,� the first place to the right of the decimal point is thirds, the next place is ninths, the next place is twenty-sevenths, etc. The value of 0.2121Base Three is

23

+ 19

+ 227

+ 181

= 54 + 9 + 6 +181

= 7081

.

Arithmetic 1. Rather than try to compute the actual value of 20084 , let�s look at some fourth powers: 14 = 1, 24 = 16, 34 = 81, 44 = 256, 54 = 625, and 64 = 1296. All of these are less than 2008, whereas 74 = 2401, which is greater than 2008. Thus there are exactly 6 positive integers less than 20084 . 2. Any number (other than zero) to the zero power is 1, so we can ignore the denominator. Evaluating the numerator, we get

213

−2

⋅ 113

3

⋅ 24

7

−1

= 73

−2

⋅ 43

3

⋅ 87

−1

= 37

2

⋅ 43

3

⋅ 78

1

= 949

⋅ 6427

⋅ 78

= 17

⋅ 43

= 421

3. There are 4 ways that 108 can be written as a sum of positive squares and positive cubes without repeats or the number 1. They are: 100 + 8, 81 + 27, 64 + 36 + 8, and 64 + 27 + 9 + 8. Algebra

1. There are eight equal intervals between 0 and 1, so the circled point must be 78

, which

is 0.875 as a decimal. The ray drawn above the number line goes to the left, which covers all the numbers less than 0.875, so the graph shows the inequality x < 0.875. 2. First let�s rewrite the equation as 4x − 5 = 34 . Now there are two possibilities: either 4x − 5 = 34 or 4x − 5 = −34 . Solving the first equation, we get 4x = 39, which

means that x = 394

. Solving the second equation, we get 4x = −29, which means that

x = −294

. The sum of these two solutions is 394

+ −294

=

104

= 212

.

3. The algebraic solution to the double inequality is shown at right. There are 28 + 1 + 34 = 63 integers

between −2817

and 3427

. (The 1 is for zero.)

−8 < 7x +1323

≤ 11

−184 < 7x +13 ≤ 253

−197 < 7x ≤ 240

−1977

< x ≤ 2407

−2817

< x ≤ 3427

m3 2008 diamond

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