MATHCOUNTS 1993-1994 State Competition Countdown Round

A man digs a hole 6 inches deep for a post to hold his mailbox A man digs a hole 6 inches deep for a post to hold his mailbox. If the square base of the post is 4 in. x 4 in., how many cubic inches of dirt will be displaced by the post? 4*4*6 = 16*6=96

96 (cubic inches)

If 2x = 8 what is 3x? 2x = 8, so x = 3. 33 = 27

27

At a certain intersection, 30% of the cars turn left, 40% turn right, and 30% go straight. What is the probability that the next car turns left and the following one turns right? (3/10)(4/10) = 12/100 = 3/25

3 25

Express in simplest form (26 + 26+ 26+ 26) 26 = 64. There are 4 of the 26 so we get 64*4 = 256. And √256 = 16

16

What is the sum of the two values that will make the following expression undefined? x – 2 x2 – 6x + 8 Factor the numerator into (x – 2)(x – 4), set it equal to zero, and solve. If x = 2 or 4, the denominator is zero, thus the fraction is undefined. The solution is 2 and 4 and the sum = 6

6 Factor the numerator into (x – 2)(x – 4), set it equal to zero, and solve. If x = 2 or 4, the denominator is zero, thus the fraction is undefined. The solution is 2 and 4 and the sum = 6

Find the largest integer less than or equal to 2-3 ·33. The product is (1/8)(27) = (27/8) which is just bigger that 3, so the answer is 3.

3

Seven equal 10-inch-long pieces are cut from an eight-foot-long piece of lumber. If the waste per cut is 1/8 inch, how many inches long is the piece that is left over? Express your answer as a mixed number. The board is 96” and you 7 cuts of 10”, so take 96 – 70 – 7(1/8)= 96 - 70 – 7/8 = 25 1/8.

25 1/8 (inches) The board is 96” and you 7 cuts of 10”, so take 96 – 70 – 7(1/8)= 96 - 70 – 7/8 = 25 1/8.

If 5 technicians can produce 64 surfboards in 4 days, how many surfboards can 8 technicians produce in 10 days? o multiply the number of surfboards produced by 8/5. Also they are working 2.5 times as many days (10 days vs. 4) so now multiply the last answer by 5/2. Or simply put 64(8/5)(5/2) = 64

256 (surfboards) Since there are 8 technicians, that means there are 8/5 as many people working, so multiply the number of surfboards produced by 8/5. Also they are working 2.5 times as many days (10 days vs. 4) so now multiply the last answer by 5/2. Or simply put 64(8/5)(5/2) = 64

What digit is used only once in the decimal representation of (1,111,111)2 Apply the pattern of multiplying by 11’s. 11*11 = 121, 111*111 = 12321. 1111*1111 = 1234321. Notice when you multiply 2 1’s (or 11) the number that appears only once is the 2, when multiplying with 3 1’s the 3 appears only once. In this problem you are multiplying 7 1’s, so the only number that will appear once is a 7.

7

The school population has increased from 3,200 to 3,600 this year The school population has increased from 3,200 to 3,600 this year. If this percent rate of increase remains constant, what will be the school population be at the end of next year? Add 3,600/8, which is 450, to 3,600 and you get 4,050.

4,050 (students)

In an arithmetic sequence the 113th term is 786 and the 125th term is 870. Find the 150th term 125-113 = 12. 870 – 786 = 84. 84/12 = 7. 870 + 25*7 = 1045.

1,045

A rectangle has length 15 inches and width 10 inches A rectangle has length 15 inches and width 10 inches. If these dimensions are both increased by 20%, by how many square inches will the area of the rectangle be increased? The new dimensions will be 18X12 which is 122 + 12*6 = 144 + 72. Take 6 away from 72 and you get the increase.

66 (square inches)

If f(x) = 2x + 3 and g(x) = 3x – 2, find f(g(f(2))) If f(x) = 2x + 3 and g(x) = 3x – 2, find f(g(f(2))) g(f(g(2))) Express your answer in the form a . b

41 31

What is the sum of the distinct prime factors of 3, 780

17

Two interior angles A and B of pentagon ABCDE are 60º and 85º Two interior angles A and B of pentagon ABCDE are 60º and 85º. Two remaining angles, C and D, are equal and the fifth angle is 15º more than twice C. Find the measure of the largest angle. The sum of the angles in a polygon is 180(n-2), so for a pentagon the interior angles add up to 540º. 540 – 145 = 395. x + x + 2x + 15 = 395. 4x = 380, x = 95. 2x + 15 is the largest angle so the answer is 205.

205 (degrees)

Simplify the following and express the result without using negative exponents. (3a-2b3c-1)2 · (√2ab-1c2) 2 3

3√2b5 2a3

Evaluate 1312 – 2(131)(31) + 312

10,000

Segment AC has three semi-circles with radii 4cm, 6cm, and 10 cm Segment AC has three semi-circles with radii 4cm, 6cm, and 10 cm. In terms of π, how many square centimeters are in the shaded area? A C

24π (sq. cm.)

Find 8*18 = 144 and 20*5 = 100, so you get 12*10 = 100

120

Find the volume, in cubic inches, of a cube whose surface area is 96 square inches.

64 (cubic inches)

Given that ab = (a2 - b2), ab express 62 as a common fraction.

8 3

A candy bar weighs 4.48 oz. If a person eats half the remaining candy bar with each bite, how many bites have been taken when there is exactly 0.07 oz remaining?

6 (bites)

Evaluate (35)(23) – (24)(34) Factor 34 and 23 from both terms you get (34* 23)(3-2) = 81*8*1= 648

648

If the sides of a right triangle are 14cm, 48 cm, and 50 cm, how many square centimeters are in its area? Do 7 * 48 = 280 + 56 = 336

336 (sq. cm.)

Central angles a, b, and c separate the circle into regions which are 1/3, 1/4, and 1/6 of the area. How many degrees are there in central angle d? a b c d Add 1/3, ¼, and 1/6 to get 9/12. The remaining sector is ¼ the circle, so the angle is 90 degrees.

90 (degrees)

What is the value of (81)0.04 • (81) 0.21 ?

3

Find the sum of all possible values for C which will make 49C345 divisible by 15. Since it ends with a 5 it’s already divisible by 5, so all you need to do is to make sure that it’s also divisible by 3. When you add the digits, they equal 25. If you add 2, 5, or 8 to 25, it will be a multiple of 3. So 2 + 5 + 8 = 15.

15

If X > 3 find Y such that Y = |X| - |3 – X| |X| - |3 – X| = | X – 3 – X| = | -3| = 3

3

How many positive integers less than 575 are divisible by 2, 3, and 5? Since the numbers must be divisible by 2, 3, and 5, they must be divisible by 30. Divide 575 by 10 you get 57.5. How many times does 3 go into 57.5??? 19 times, so there are 19 integers that work.

19 (integers)

One bus company has a bus arrive at the Parkview bus stop every 16 minutes, while another bus company has a bus arrive at the same stop every 20 minutes. If a bus from each company arrives at Parkview bus stop simultaneously at 3:00 PM, what is the next time they will arrive there simultaneously? The GCF of 16 and 20 is 80, so 80 minutes after 3:00 they will meet again. Thus, they meet at 4:20 PM.

4:20 (PM) or 16:20

Find the value of (n + 1)! when n = 100 (n – 1)! (101)!/(99)!= 101*100 = 10,100

10,100

Find the value of x in the triangle shown Find the value of x in the triangle shown. Express your answer in simplest radical form. 45° 16 x x

8√2 (sq. cm.)

Express 0.003 X 0.00033 in scientific notation.

9.9 X 10-7

A girls basketball team has 4 players of heights 6’1”, 5’9”, 5’8”, and 5’11”. If the average height of the 5-girl team is 5’10”, then the fifth player must be 5’a” tall. Find a.

9

The sum of two numbers is 19 and their difference is 5 The sum of two numbers is 19 and their difference is 5. What is their product?

84

Express as a common fraction.

- 3 2

What is the greatest common factor of (a2 – 9) and (3a2 – 3a – 36)?

( a + 3)

What is the value of x in the diagram? x 2 60° 45°

√6 (units)

Find the area of the shaded cross section of the magnet, where OE = 2, OF = 4, OP = 4, and both arcs shown are semicircles. Leave your answer in terms of π. E 2 F 2 o 4 2 P 2

16 + 6π (sq. units)

Find the product of (1/4)-3(8)-2

1

What is the value of 6! (6 – 4)!4!

15

If a light bulb has an expected life of 1,000 hours, for how many complete days of continuous use can it be expected to work? 1000/25 = 40, so since there is one less hour that you are dividing by, 24 instead of 25, you can expect that your final answer will be larger than 40. The actual answer is 41.66666… so 41 days.

41 (days)

Laws in certain states require a 60% approval rate from voters before an issue is “passed.” If 1,140 people voted in favor of the law, and this represents 57% of the total number of voters, how many voters were there in this election? 114/57 = 2, add 3 decimal places, one for the 114 and two for the 57%, so there were 2,000 voters.

2,000 (voters)

If a 25cm X 12cm X 7cm full box of paper towels contains 175 individual towels, what is the volume of each towel in cm3? 125*7 = 175, so the area of each paper towel is 12cm3.

12 (cm.3)

Given the square of an integer x is 1,521 Given the square of an integer x is 1,521. What is the value of (x + 1)(x – 1) X2 = 1521 and (x + 1)(x – 1) = X2 – 1, so (x + 1)(x – 1) = 1520

1,520

A step pyramid is formed by stacking successively smaller “steps” as illustrated in the drawing. If the edges of each square “step” have lengths 9,7,5,3, and 1, and each “step” is 1 unit high, find the number of cubic units in the volume of the pyramid. 1 3 3 5 5 7 7 9 9

165 (units3)

What is the value of (3/5)2 + 2(3/5)(2/5) + (2/5)2? This is the factored form of a perfect square trinomial. (a + b)2 = a2+ 2ab + b2. So you simply take the two fractions, (3/5 + 2/3)2 = 1.

1

The configuration below is built from 27 segments The configuration below is built from 27 segments. How many paths are there from A to B using exactly five of these segments? B A You must use the diagonals, otherwise it takes 6 segments. There are tone path each going up to the diagonal and to the right from A. There are four paths going to the one in the middle.(up-right-diagonal-up-right) or any combination of up and right.

6 (paths)

Suppose the estimated $45 billion cost of sending a person to Mars is shared equally by the 250 million people of the United States. What is each person’s share? The problem is reduced to 45/25 = 1 20/45 = 1.8 but since there is one more zero in 45 billion than 250 million, add a zero onto the 1.8 for $180 per person.

180 (dollars)

If (x + 1)1/2 = 21/3 find the value of (x + 1)2 in simplest radical form. If you raise (x + 1)1/2 to the 4th power you get (x + 1)2. This means that you must also raise 21/3 to the 4th power and you get 24/3. Which is 23/3 * 21/3 = 2 * cube root of 2.

2 ·3√2

How many numbers can be written as the sum of two or more distinct members of the set {0, 1, 2, 3, 4, 5} Using 0 and another number you get sums of 1, 2, 3, 4, and 5 for a total of 5 different numbers. Using 1 with another number you only add one new sum, 1 + 5, or 6, for a new total of 6. The same thing happens when you use 2, 3, … and another number. 5C2 = 10, (more to come…)

15 (numbers)

How many positive integers less than 100 can be expressed as a power of 3? 30, 31, 32, 33, 34

5 (integers)

How many ways can 5 distinct paperback books and 1 hardcover book be arranged on a shelf if the hardcover book must be the rightmost book on the shelf? Multiplication Counting Principle: 5*4*3*2*1 or 5! = 120.

120 (ways)

Find the negative root(s) of the equation (x + 1)2 = 2(x + 1) On the left side we get x2 + 2x + 1 = 2x + 2. The 2x’s cancel from both sides, leaving x2 = 1, so x = +/- 1. The negative root is –1.

- 1

Given x varies directly as y2 and x = 3 when y = 5, find x when y = 6 Given x varies directly as y2 and x = 3 when y = 5, find x when y = 6. Express your answer as a common fraction. With direct variation you divide x/y2, so you get 3/25 = x/36 and multiply by 36 to get 108/25.

108 25

17 (yards)

√2

54 (diagonals)

2

29 ¼

51

24 (sq. cm.)

18

1 38

1

34 27

7