AMC 10 Basic 1 2014 pratice paper

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

1. 1 point Tom's age is years, which is also the sum of the ages of his three

children. His age years ago was twice the sum of their ages then. What is

?

Solution: (D)

Tom's age years ago was . The sum of his three children's ages

at that time was . Therefore , so and

. The conditions of the problem can be met, for example, if Tom's

age is and the ages of his children are , , and . In that case

and .

2. 1 point Yan is somewhere between his home and the stadium. To get to the

stadium he can walk directly to the stadium, or else he can walk home and then

ride his bicycle to the stadium. He rides times as fast as he walks, and both

choices require the same amount of time. What is the ratio of Yan's distance

from his home to his distance from the stadium?

Solution: (B)

Let be Yan's walking speed, and let and be the distances from Yan

to his home and to the stadium, respectively. The time required for Yan to

walk to the stadium is , and the time required for him to walk home is

. Because he rides his bicycle at a speed of , the time required for

him to ride his bicycle from his home to the stadium is . Thus

As a consequence, , so .

The required ratio is .

OR

Created using

Copyright (c) 2013, Edfinity. Reproduction without Edfinity's permission is strictly forbidden.

Page 1 of 20

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Because we are interested only in the ratio of the distances, we may assume

that the distance from Yan's home to the stadium is mile. Let be his

present distance from his home. Imagine that Yan has a twin, Nay. While

Yan walks to the stadium, Nay walks to their home and continues of

a mile past their home. Because walking of a mile requires the same

amount of time as riding mile, Yan and Nay will complete their trips at

the same time. Yan has walked miles while Nay has walked

miles, so . Thus , , and the required ratio

is .

3. 1 point Josh and Mike live 13 miles apart. Yesterday Josh started to ride his

bicycle toward Mike's house. A little later Mike started to ride his bicycle

toward Josh's house. When they met, Josh had ridden for twice the length of

time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden

when they met?

Solution: (B)

Because , the distance Josh rode was

of the distance that Mike rode. Suppose Mike rode miles.

Then the number of miles between their houses is

Thus .

4. 1 point The ratio of the radius of the sector to that of the inscribed circle in the

picture is . What is the ratio of their areas?

Created using


Page 2 of 20

(A) (B) (C) (D) (E)

Solution: (D)

Step 1: Understand the problem

The diagram makes the question clear, even if it isn't obvious how to use the

given information to answer it.

Step 2: Devise a plan

The only information you have to work with involves the radii of the sector

and the inscribed circle, so name the unknowns by letting and be the

radii of the sector and the circle, respectively. Then take a possible step and

add some radii to the picture. Let be the center of the sector, let be the

center of the circle, and let the circle be tangent to the arc of the sector at

. Then and are radii of the sector and the circle, respectively. It

appears that lies on , so make a conjecture that this is the case. After

you draw , it might occur to you to draw a radius from to one of the

other points of tangency, call it , in order to create right in the

picture. You can probably express the side lengths of the triangle in terms

of and , but what then? You will need to know the central angle of the

sector in order to answer the question. Again, take a possible step and see

what you can learn from .

Created using


Page 3 of 20

(A) (B) (C) (D) (E)

Step 3: Carry out the plan

Because you are concerned only with ratios, you can choose convenientnumbers and assume that and . To verify that lies on

, look for symmetry. If you draw a radius from to the remaining point

of tangency, call it , then and are congruent. It follows

that bisects . Furthermore, the equal angles and

divide the sector into two congruent sectors, so radius bisects .

Therefore does lie on .

Now focus on . You know that the length of one leg is , and

the length of the hypotenuse is . Therefore

is a right triangle, and . That means that

the angle of the sector is radians, and the area of the

sector is . The area of the circle is , so the ratio of

the two areas is .

5. 1 point Points and are both on the line segment and on the same side

of its midpoint. divides in the ratio , and divides in the ratio

. If , then

Solution: (C)

Suppose that and are both closer to than to . Then is of

the way from to , and is of the way from to . Thus the

fraction of segment that is occupied by is . It follows

that .

6. 1 point Two bottles of equal volume contain both water and juice. The ratios of

the volume of water to the volume of juice in the two bottles are, respectively,

and . The contents of the two bottles are put into one big bottle. What

is the ratio of the volume of water to the volume of juice in this bottle?

Created using


Page 4 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

Solution: (A)

Because the question is concerned only with ratios of volumes, it is

permissible to choose convenient volumes. Suppose each of the original

bottles contains a total of liters of liquid. Then the first bottle contains

liters of water and liters of juice, and the second bottle contains liters of

water and liter of juice. The big bottle contains liters of water

and liters of juice, so the ratio of water to juice in the big bottle is

.

7. 1 point The value of is closest to which of the

following numbers?

Solution: (C)

The value of the expression is approximately

.

Note: Rounded to two decimal places, the expression has the value .

8. 1 point What is

Solution: (C)

The given expression is equal to

Created using


Page 5 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

9. 1 point All the students in Deshawn's class were given a lengthy reading

assignment. By Monday, only of the students had completed the

assignment, but by Friday of the students had completed it. Of the students

who had not completed the assignment by Monday, what percent had completed

it by Friday?

Solution: (D)

Note that percent of the class had not completed the

assignment by Monday and percent of the class completed the

assignment between Monday and Friday. The latter group is

percent of the former.

10. 1 point Ali says that of his books are novels, and of them are poetry.

Given that he has between and books, how many books does he have?

Solution: (D)

Let be the number of books that Ali has. Because of his books

are novels, must be a multiple of , and because of his books are poetry,

must be a multiple of . The least common multiple of and is , so

must be a multiple of . The only multiple of between and is ,

so Ali has books.

11. 1 point The average of different positive integers is . What is the largest

possible value that one of these integers could have?

Solution: (C)

Created using


Page 6 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

The sum of the integers is . To get the largest possible

value for the largest of the integers, choose the smallest possible values for

the other . Therefore the largest possible value of the largest integer is

.

12. 1 point What number is halfway between and ?

Solution: (D)

The number that is halfway between and is their average, which

is .

13. 1 point In a certain city, the streets that cross Main Street are numbered

consecutively, and the distance along Main Street between consecutive cross

streets is constant. Jack and Jill begin to walk along Main Street at the corner of

First Street. Jack walks to the corner of Third Street, and Jill continues walking

to the corner of Seventh Street. Jill walks how many times as far as Jack?

Solution: (D)

Jack walks blocks, and Jill walks blocks, so Jill walks

times as far as Jack.

14. 1 point Weifeng writes down consecutive positive integers. If the smallest

and the largest numbers are perfect squares, what is the sum of the digits in the

smallest number she writes down?

Solution: (A)

Created using


Page 7 of 20

(A) (B) (C) (D) (E) All of these are possible.

Let the smallest and the largest numbers Weifeng writes down be and

respectively. Since they are the ends of a block of consecutive numbers,

. The ways of expressing as a product

of two positive integers are , and . Only when

and are and both integers. In this case , so the first

number Weifeng writes down is . The sum of the digits is .

15. 1 point In the Stanley Cup Championship, seven games are scheduled. Games

, , and are to be played in the arena of one team, while Games , and

are to be played in the arena of the other team. However, the Championship is

awarded as soon as either team wins four games. That is, one or more of Games

, , and may not be played. If exactly games are won by the home team,

then the total number of games cannot be

Solution: (C)

Suppose Team A hosts Games , , , and and Team B hosts Games , ,

and . The following examples show that it is possible for the total number

of games to be , , or :

If Team A wins the first games, then the total number of games is , and

the home team has won Games and .

If Team A wins Games , , , and , then the total number of games is ,

and the home team has won Games and .

If Team B wins Games , , , and , then the total number of games is ,

and the home team has won games and .

To show that it is not possible for the total number of games to be , consider

the following cases:

Created using


Page 8 of 20

(A) (B) (C) (D) (E)

If each team wins home game, then each team loses home games, and it

follows that each team also wins away games. Thus each team has won

games, and Game must be played.

If either team wins home games, then that team must also win all of its

away games. Because the first games include away games for each team,

the winning team will have won a total of games, a contradiction.

16. 1 point Lines parallel to the base divide each of the other two sides of the

triangle shown into equal segments. Which percentage of the area of triangle

is grey?

Solution: (C)

Created using


Page 9 of 20

(A) (B) (C) (D) (E)

It may be assumed that the large triangle has area . Let , , and be the

top, bottom left, and bottom right vertices of the large triangle, respectively.

Label the points on the left side from top to

bottom, and label the points on the right side

from top to bottom. For , is similar to with

, so the area of is . Therefore the total grey area is

which is of the area of .

17. 1 point A student recorded the exact percentage frequency distribution for a

set of measurements. However, the student neglected to indicate , the total

number of measurements. What is the smallest possible value of ?

Solution: (B)

Since is , must be at least . Since and

, is the answer.

Created using


Page 10 of 20

(A) (B) (C) (D) (E)

18. 1 point A snail is moving directly away from an anthill at cm/s. When the snail

is meter away, an ant leaves the anthill, runs to the snail, then runs back to the

anthill at a speed of cm/s. The ant then runs to the snail and back to the anthill

at cm/s. The ant keeps running back and forth between the anthill and the snail,

increasing its speed by cm/s each time it leaves the anthill. When the ant has

run back and forth times, how many meters from the anthill is the snail?

Solution: (D)

Suppose the snail is meters away at the start of the ant's round

trip, so , and the ant takes seconds to catch

up to the snail and takes the same amount of time to run back to the

anthill. Thus during the ant's trip, the snail moves a distance of

meters. Therefore ,

and it follows that the snail's distance from the anthill immediately before

the ant's trip is

meters.

19. 1 point The numbers , , , and are to be placed in the remaining spaces in

the H-shaped figure below, one number per space, such that the sum of the three

numbers along each of the three lines of the letter H is the same. The number of

ways the spaces can be filled is

Created using


Page 11 of 20

(A) (B) (C) (D) (E)

(A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 2005

(D) 2005 and 2006 (E) 2006 and 2007

Solution: (E)

Let be the number immediately below the . Then the numbers and

appear in two of the three sums, and all the other numbers appear in

one. Therefore the sum of the three sums is

. Because the three sums

are equal, must be a multiple of , implying that is either or .

If , then the sum of the three sums is , so each sum is . Thus the

number below the must be , and the number to the right of the must be

. The remaining numbers, and , can be placed in the remaining positions

in either of ways, so there are ways of filling the spaces in this case.

If , then the sum of the three sums is , so each sum is . Thus the

number below the must be , and the number to the right of the must be

. The remaining numbers, and , can be placed in the remaining positions

in either of ways, so there are ways of filling the spaces in this case.

Thus the spaces can be filled in a total of ways.

20. 1 point At Euclid High School, the number of students taking the AMC10 was

in 2002, in 2003, in 2004, in 2005, and in 2006, and is in 2007.

Between what two consecutive years was there the largest percentage increase?

Solution: (A)

Created using


Page 12 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

Between 2002 and 2003, the increase was Between the

other four pairs of consecutive years, the increases were

Therefore the largest percentage increase occurred

between 2002 and 2003.

21. 1 point Last year Mr. John Q. Public received an inheritance. He paid

in federal taxes on the inheritance, and paid of what he had left in state

taxes. He paid a total of for both taxes. How many dollars was the

inheritance?

Solution: (D)

After paying the federal taxes, Mr. Public had of his inheritance money

left. He paid of that, or of his inheritance, in state taxes. Hence

his total tax bill was of his inheritance, and his inheritance was

.

22. 1 point Each day Walter gets for doing his chores or for doing them

exceptionally well. After days of doing his chores daily, Walter has received

a total of . On how many days did Walter do them exceptionally well?

Solution: (A)

Walter gets an extra per day for doing chores exceptionally well. If he

never did them exceptionally well, he would get for days of chores.

The extra must be for days of exceptional work.

Created using


Page 13 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

23. 1 point One ticket to a show costs at full price. Susan buys tickets using

a coupon that gives her a discount. Pam buys tickets using a coupon that

gives her a discount. How many more dollars does Pam pay than Susan?

Solution: (C)

Susan pays dollars. Pam pays

dollars, so she pays more dollars than Susan.

24. 1 point Isabella's house has bedrooms. Each bedroom is feet long,

feet wide, and feet high. Isabella must paint the walls of all the bedrooms.

Doorways and windows, which will not be painted, occupy square feet in

each bedroom. How many square feet of walls must be painted?

Solution: (E)

The perimeter of each bedroom is feet, so the surface to be

painted in each bedroom has an area of square feet. Since

there are bedrooms, Isabella must paint square feet.

25. 1 point Mary is older than Sally, and Sally is younger than Danielle.

The sum of their ages is years. How old will Mary be on her next birthday?

Solution: (B)

Let Danielle be years old. Sally is younger, so she is years old.

Mary is older than Sally, so Mary is years old. The

sum of their ages is years, so .

Therefore Mary's age is years, and she will be on her next

birthday.

Created using


Page 14 of 20

(A) 2 (B) 4 (C) 10 (D) 20 (E) 40

(A) 5 (B) 10 (C) 15 (D) 20 (E) 25

(A) (B) (C) (D) (E)

26. 1 point A positive number has the property that of is . What is ?

Solution: (D)

We have Because , it follows that

.

27. 1 point Mary is about to pay for five items at the grocery store. The prices of the

items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-

dollar bill. Which of the following is closest to the percentage of the $20.00 that

she will receive in change?

Solution: (A)

The five items cost approximately dollars, so

Mary's change is about \ 5$ percent of her $20.00.

28. 1 point The sum of the digits of a two-digit number is subtracted from the

number. The units digit of the result is . How many two-digit numbers have

this property?

Solution: (D)

Let be the two-digit number. When is subtracted the result is

. The only two-digit multiple of that ends in is , so .

The ten numbers between and , inclusive, have this property.

29. 1 point In how many zeros does end?

Solution:

Created using


Page 15 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

The problem is to find the highest power of which divides . Since

divides more often than does, the highest power of dividing

is equal to the highest power of dividing . There is a factor

for every multiple of not exceeding (there are such multiples),

an additional for every multiple of ( of them), yet another for every

multiple of ( of them), yet another for every multiple of ( of

them) and another for every multiple of ( of them). Therefore, the

number of zeros is

In general, the number of zeros in which ends with is given by

30. 1 point How many primes less than have as the ones digit? (Assume the

usual base representation.)

Solution: (C)

, , , , and are primes; for instance, for it is sufficient to

check that it is not divisible by , , and , which are the only primes less

than its square root. On the other hand, , , and

are not primes.

31. 1 point A college student drove his compact car miles home for the weekend

and averaged miles per gallon. On the return trip the student drove his

parents' SUV and averaged only miles per gallon. What was the average gas

mileage, in miles per gallon, for the round trip?

Solution: (B)

Created using


Page 16 of 20

(A) (B) (C) (D) (E)

(A) (B) (C) (D) (E)

The student used gallons on the trip home and

gallons on the trip back to school. So the average gas mileage for the round

trip was

32. 1 point The 2007 AMC 10 will be scored by awarding points for each correct

response, points for each incorrect response, and points for each problem

left unanswered. After looking over the problems, Sarah has decided to

attempt the first and leave only the last unanswered. How many of the first

problems must she solve correctly in order to score at least points?

Solution: (D)

Sarah will receive points for the three questions she leaves unanswered,

so she must earn at least points on the first problems.

Because she must solve at least of the first problems

correctly. This would give her a score of .

33. 1 point Tom's age is years, which is also the sum of the ages of his three

children. His age years ago was twice the sum of their ages then. What is

?

Solution: (D)

Tom's age years ago was . The sum of his three children's ages

at that time was . Therefore , so and

. The conditions of the problem can be met, for example, if Tom's

age is and the ages of his children are , , and . In that case

and .

Created using


Page 17 of 20

(A) (B) (C) (D) (E)

(A) (B) and seconds (C) (D)

(E)

34. 1 point A gallon of paint is used to paint a room. One third of the paintis used on

the first day. On the second day, one third of the remaining paint is used. What

fraction of the original amount of paint is available to use on the third day?

Solution: (D)

After the first day, of the paint remains. On the second day,

of the paint is used. So for the third day of the

original gallon of paint is available.

35. 1 point Anna and Beth begin to run around a circular track at noon, starting at

the same point and running in the same direction. Anna completes a lap every

seconds, and Beth completes a lap every seconds. At what time after noon

do Anna and Beth first reach their starting point simultaneously?

Solution: (C)

The least common multiple of and is , and seconds is equal

to minutes. Therefore Anna and Beth first reach their starting point

simultaneously at .

36. 1 point Jimmy runs a successful pizza shop. In the middle of a busy day, he

realizes that he is running low on ingredients. Each pizza must have lb of

dough, lb of cheese, lb of sauce, and lb of toppings, which include

pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently has

lbs of dough, lbs of cheese, lbs of sauce, lbs of pepperonis, lbs

of mushrooms, lbs of olives, and lbs of sausages, what is the maximum

number of pizzas that Jimmy can make?

Created using


Page 18 of 20

Solution:

The limiting factor is the cheese. With only lbs of cheese, the most pizzas

that can be made is 80.

37. 1 point In a non-recent edition of it was stated

that the number

is a number, that is, if multiplied by any positive integer the resulting

number always contains the ten digits in some order with possible

repetitions.

a. Prove or disprove the above statement.

Solution:

The statement is false. Observe that

Thus

which does not contain all the ten digits.

b. Are there any persistent numbers smaller than the above number?

Solution:

There are no persistent numbers! Our proof is indirect. Assume that

is persistent. We can express in the form where is

relatively prime to both and . Since

must also be persistent, all multiples of must contain the nine

nonzero digits. We now invoke Euler's generalization of Fermat's

Theorem: If is relatively prime to then

Created using


Page 19 of 20

where (Euler's -function) is the number of positive integers less

than or equal to that are relatively prime to }. Now we have

and this gives a contradiction, for should contain all nine nonzero

digits, but the left side of contains only nines.

For an alternative, more elementary, solution, assume again that

is persistent and consider the remainders obtained by dividing the

following numbers by :

where the last number has digits. Since at most different

nonzero remainders can result, either one of the above numbers is

divisible by , in which case is not persistent, or else two of them,

say

give the same remainder, in which case their difference

is divisible by , and is not persistent.

Created using


Page 20 of 20

AMC 10 Basic 1 2014 pratice paper

Documents