Tuesday, April 24, 2018

Individual Contest

FORM A

Do Not Open This Booklet Until Instructed To Do So By The Proctor

1. How many two-digit numbers are divisible by 2, 4, 6 and 8? A. 5 B. 4 C. 3 D. 2 E. 1 F. 6

2. John was asked by his teacher to subtract 3 from a certain number and then divide the result by 9.

Instead, he subtracted 9 and then divided the result by 3, obtaining an answer of 43.What would his answer have been if he had worked the problem correctly? A. 15 B. 135 C. 258 D. 45 E. 138 F. 129

3. In the expression !!+ !

!+ !

!, each letter is replaced by a different digit from 1, 2, 3, 4, 5 and 6. What is

the largest possible value of this expression?

A. 8 !! B. 10 !

! C. 9 !

! D. 10 !

! E. 9 !

! F. 9 !

!

4. Let 𝑥 = 2!"# ∙ 3!" ∙ 5!"#, 𝑦 = 2!"# ∙ 3!" ∙ 5!"#, and 𝑧 = 2!"# ∙ 3!" ∙ 5!"#. What is the order of the three numbers from smallest to largest?

A. x,y,z B. y,z,x C. z,x,y D. x,z,y E. y,x,z F. z,y,x

5. The increasing list of five different integers {3, 4, 5, 8, 9} has a sum of 29. How many increasing lists

of five different single-digit positive integers have a sum of 33?

A. 2 B. 4 C. 3 D. 1 E. 5 F. 6

6. How many two-digit numbers exist such that when the product of the digits is added to the sum of the digits, the result is equal to the original two-digit number?

A. 20 B. 7 C. 14 D. 9 E. 2 F. 10

7. Three different views of the same cube colored with six different colors are shown below. What color is on the face opposite the orange face? A. Red B. Orange C. Blue D. Violet E. Yellow F. Green

red

yellowgreen red

blue

orange

blue

greenviolet

8. At a gathering of professionals, all but 30 were musicians; all but 40 were pediatricians; and all but 50 were journalists. Each person has exactly one profession. If only musicians, journalists, and pediatricians attend this gathering, how many journalists were present? A. 20 B. 10 C. 30 D. 70 E. 50 F. 60

9. Two numbers are chosen without replacement from the integers 1, 2, 3, 4, 5, 6, 7, 8. What is the probability that the sum is 9?

A. !! B. !

! C. !

! D. !

! E. !

! F. !

!

10. A gumball machine has only 4 gumballs left. Three of them are green and one is red. You put a coin

in the machine, turn the crank, and two gumballs come out. What is the probability that both gumballs are green?

A. !! B. !

! C. !

! D. !

! E. !

! F. !

!

11. Two squares are positioned as shown below. The smaller square has side length 1 and the larger square has side length 7.Find the length of AB.

A. 72 B. 85 C. 10 D. 8 E. 113 F. 9

12. A thin belt is stretched around three pulleys, each of which is 2 feet in diameter. (See diagram above).

The distances between the centers of the pulleys are 6 feet, 9 feet, and 13 feet. To the nearest 0.1 foot, how long is the belt?

A. 34.3 B. 31.0 C. 30.0 D. 37.4 E. 46.8 F. 34.0

13. What is the measure of the smaller angle formed by the hour hand and minute hand of a circular clock

at 2:46 ?

A. 84° B. 137° C. 167° D. 154° E. 139° F. 172° 14. What is the maximum number of points of intersection when two circles and five straight lines

intersect?

A. 10 B. 20 C. 26 D. 30 E. 32 F. 21

B

A

15. A jar contains only red and yellow jelly beans. If a child eats 1 red jelly bean, 1/7 of the remaining candies will be red. If instead the child eats 5 yellow jelly beans, 1/6 of the remaining candies will be red. How many jelly beans are in the jar? A. 29 B. 43 C. 37 D. 113 E. 71 F. 47

16. Below is shown a growing sequence of shapes made out of toothpicks. The first shape is made using 4 toothpicks, the second uses 12 toothpicks and the third uses 24. How many toothpicks will there be in the 60th shape?

A. 7,320 B. 3,600 C. 480 D. 14,400 E. 4,800 F. 5,480

17. Eric the Sheep is at the end of a line of 50 sheep waiting to be shorn. But being an impatient sort of

sheep, Eric sneaks up the line two places every time the sheerer takes a sheep from the front to be shorn. So, for example, while the first sheep is being shorn, Eric moves ahead so that there are two sheep behind him in line. If at some point it is possible for Eric to move only one place, he does that instead of moving ahead two places. How many sheep get shorn before Eric?

A. 25 B. 19 C. 17 D. 21 E. 23 F. 15

18. My grandfather’s papers included an old receipt: 72 turkeys purchased for $ X67.9Y

The first and last digits (shown here as X and Y) were smudged and illegible. What is the value of X ? A. 8 B. 5 C. 6 D. 9 E. 7 F. 3

19. A toothpaste tube holds 144 cubic cm of toothpaste. The nozzle of the tube is round with a diameter of 0.75 cm.Assume that the toothpaste is deposited on the counter in a straight line such that the diameter is the same as that of the nozzle (0.75 cm). Approximately how long a line (in cm) of toothpaste can be made with a full tube of toothpaste? A. 64 B. 256 C. 61 D. 108 E. 326 F. 192

20. When Elizabeth visits her friend Andrew, she must first walk uphill, then walk downhill. She walks

home by the same route. She always walks 2 km/h when going uphill, and 6 km/h when going downhill. If her total walking time is 1 hour, what is the total round-trip distance she walks (in km) ?

A. !! B. 8 C. 3 D. 4.5 E. 4 F. 3.5