Spring Level 5 | 07/31/2010

(1) Trapezoid ABCD has vertices A(1; 0), B(0; 4), C(m; 4) and D(k; 0), with

m > 0 and k > 0. The line y = x + 4 is perpendicular to the line containing side CD, and

the area of trapezoid ABCD is 34 square units. What is the value of k?

(2) What integer n has the property that 5

96

greater than n

72

and 5

96

is less

than (n + 1)

72

?

(3) If 74 hens lay 74 dozen eggs in 74 days and if 37 hens eat 37 kilograms of

wheat in 37 days, how many kilograms of wheat are needed to produce 1 dozen eggs?

(4) On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one

letter per magnet. Two vowels and three consonants fall o and are put away in a bag. If

the Ts are indistinguishable, how many distinct possible collections of letters could be put

in the bag?

(5) From a circular piece of paper with radius BC, Je removes the unshaded

sector shown. Using the larger shaded sector, he joins edge BC to edge BA (without

overlap) to form a cone of radius 12 centimeters and of volume 432 cubic centimeters.

What is the number of degrees in the measure of angle ABC of the sector that is not

used?

C

A

B

(6) In the gure below, the shaded region is formed by drawing two parallel

segments which connect the midpoints of congruent squares. Each square has side length 1

centimeter. What is the number of square centimeters in the shaded region? Express your

answer as a common fraction.

(7) In how many ways can 4 on/o switches in a row be set so that no two

adjacent switches are on?

(8) What is the sum of the numbers less than 200 that have exactly 9 divisors?

(9) How many integers n satisfy the condition 100 < n < 200 and the condition

n has the same remainder whether it is divided by 6 or by 8?

(10) How many distinct pairs of perfect squares dier by 35? (The pair a; b is the

same as the pair b; a.)

(11) What is the greatest integer value of x such that

x

2

+2x+5

x3

is an integer?

(12) Your family traveled 260 miles to the beach at an average speed of 65 mph.

Trac on the return trip slowed your speed to an average of 40 mph. To the nearest whole

number, what was the mean speed for the entire trip? Express your answer in miles per

hour.

(13) A rectangular block of candy 10 inches by 10 inches by 5 inches is coated on

all faces with a very thin layer of chocolate. The block of candy is then cut into cubes

measuring 1 inch by 1 inch. What percent of the cubes have no chocolate on them?

Express your answer to the nearest tenth.

(14) Alyssa rode her bicycle for seven hours. She started on level ground and

rode at a rate of 8 mph. She came to a hill that slowed her down to a rate of 5 mph. Upon

reaching the top of the hill, she turned around and descended at a rate of 20 mph. Finally,

she returned home on the at portion at 8 mph. What is the total number of miles that

she rode?

(15) Three people, Alan, Beth, and Cindy, pooled their money to purchase lottery

tickets. Alan invested $25:00. Beth invested $20:00 and Cindy invested $35:00. Their

winnings totaled 6:4 million dollars. How many dollars was Beth's share of the winnings if

the amount received was proportional to the amount invested?

(16) Two boards, one four inches wide and the other six inches wide, are nailed

together to form an X. The angle at which they cross is 60 degrees. If this structure is

painted and the boards are separated what is the area of the unpainted region on the

four-inch board? (The holes caused by the nails are negligible.) Express your answer in

simplest radical form.

60

4

6

(17) Three positive integers a, b, and c satisfy a b c = 8! and a < b < c . What

is the smallest possible value of c a?

(18) The time is 9:00. On a 12-hour clock, how many minutes elapse before the

hour hand and minute hand are 28 degrees apart for the rst time?

(19) The diagonals of a rectangle intersect at point P . Point P is 5 centimeters

further from the shorter side than the longer side. The perimeter of the rectangle is 44

centimeters. What is the number of square centimeters in the area of the rectangle?

(20) Je will pick a card at random from ten cards numbered 1 through 10. The

number on this card will indicate his starting point on the number line shown below. He will

then spin the fair spinner shown below (which has three congruent sectors) and follow the

instruction indicated by his spin. From this new point he will spin the spinner again and

follow the resulting instruction. What is the probability that he ends up at a multiple of 3

on the number line? Express your answer as a common fraction.

0 1 2 3 4 5 6 7 8 9 10 11 12-1

move

1 space

left

move

1 space

right

move

1 space

right

Copyright MATHCOUNTS Inc. All rights reserved

Answer Sheet

Number Answer Problem ID

1 6 1A02

2 8 22531

3 2 C35A1

4 75 collections 045

5 72 degrees 10D52

6 5=4 sq. cm 11A01

7 8 34141

8 332 B5B11

9 25 integers 2A02

10 2 pairs 00C1

11 23 31B4

12 50 54141

13 38.4 percent AB3C

14 56 05B01

15 1600000 dollars BB121

16 16

p

3 sq inches C45

17 4 5C552

18 44 minutes AD02

19 96 square centimeters 1B121

20 31/90 00D52

Copyright MATHCOUNTS Inc. All rights reserved

Solutions

(1) 6 ID: [1A02]

The line y = x + 4 has slope 1. Since line CD is perpendicular to this, it has slope

1

1

= 1. Thus,

4 0

m k

= 1) m k = 4:

Trapezoid ABCD has height 4 and bases BC = m and AD = k + 1. The area of the

trapezoid is thus

1

2

(4)(m + k + 1). Setting this equal to 34 gives

1

2

(4)(m + k + 1) = 34) m + k = 16:

Subtracting m k = 4 from m + k = 16, we get 2k = 12, or k = 6 .

(2) 8 ID: [22531]

We have n

72

< 5

96

< (n + 1)

72

. Taking each expression to the

1

24

power, we get

n

3

< 5

4

= 625 < (n + 1)

3

. Notice that 8

3

= 512, and 9

3

= 729, so n = 8 .

(3) 2 ID: [C35A1]

No solution is available at this time.

(4) 75 collections ID: [045]

Let's divide the problem into two cases: one where 0 or 1 T's fall o and one where both

T's fall o:

0 or 1 T's:(3

2

)(6

3

)= 3 20 = 60

2 T's:(3

2

)(5

1

)= 3 5 = 15

Total: 60 + 15 = 75

(5) 72 degrees ID: [10D52]

Solving

1

3

(12 cm)

2

(h) = 432 cm

3

, we nd that the height h of the cone is 9 cm. Since

the radius is 12 cm and the height is 9 cm, the slant height of the cone, which is the same

as the distance from B to C, is

p

9

2

+ 12

2

= 15 centimeters. The length of major arc AC

is equal to the circumference of the cone, which is 2(12 cm) = 24 cm. The distance all

the way around the circle is 2(BC) = 30 cm. Therefore, the central angle of major arc

AC measures

(24 cm

30 cm

)360

= 288 degrees. The measure of angle ABC is

360

288

= 72 degrees.

(6) 5=4 sq. cm ID: [11A01]

No solution is available at this time.

(7) 8 ID: [34141]

No solution is available at this time.

(8) 332 ID: [B5B11]

No solution is available at this time.

(9) 25 integers ID: [2A02]

Since n has the same remainder whether divided by 6 or by 8, we can write that

n = 6a + r = 8b + r , where 0 r 5. This implies that 3a = 4b, and so a is a multiple of

4 and we can write a = 4k for some integer k . Since 100 < n < 200, we see that

95 < 6a < 200, or

95

24

< k