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USA

AMC 10 2014

A

1 2 1 5 1 ?1 ? 10 25 (C) 2

1 What is 10 · (A) 3

+

+

(B) 8

(D)

170 3

(E) 170

1 2 Roy’s cat eats 1 3 of a can of cat food every morning and 4 of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat ?nish eating all the cat food in the box?

(A) Tuesday

(B) Wednesday

(C) Thursday

(D) Friday

(E) Saturday

3 Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $2.50 each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $0.75 for her to make. In dollars, what is her pro?t for the day? (A) 24 (B) 36 (C) 44 (D) 48 (E) 52

4 Walking down Jane Street, Ralph passed four houses in a row, each painted a di?erent color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

5 On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest scored 100 points. What is the di?erence between the mean and median score of the students’ scores on this quiz? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

6 Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days? (A)

bde ac

(B)

ac bde

(C)

abde c

(D)

bcde a

(E)

abc de

7 Nonzero real numbers x, y , a, and b satisfy x < a and y < b. How many of the following inequalities must be true? (I) x + y < a + b (II) x ? y < a ? b (III) xy < ab (IV) (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 18!19! 2

x y

<

a b

8 Which of the following number is a perfect square? 14!15! 15!16! 16!17! 17!18! (A) (B) (C) (D) 2 2 2 2

(E)

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 1

USA

AMC 10 2014

√ 9 The two legs of a right triangle, which are altitudes, have lengths 2 3 and 6. How long is the third altitude of the triangle? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

10 Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that start with b? (A) a + 3 (B) a + 4 (C) a + 5 (D) a + 6 (E) a + 7

11 A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10% o? the listed price if the listed price is at least $50 Coupon 2: $20 o? the listed price if the listed price is at least $100 Coupon 3: 18% o? the amount by which the listed price exceeds $100 For which of the following listed prices will coupon 1 o?er a greater price reduction than either coupon 2 or coupon 3? (A) $179.95 (B) $199.95 (C) $219.95 (D) $239.95 (E) $259.95

12 A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

√ (A) 27 3?9π

√ (B) 27 3?6π

√ (C) 54 3?18π

√ (D) 54 3?12π

√ (E) 108 3?9π

13 Equilateral ABC has side length 1, and squares ABDE , BCHI , CAF G lie outside the triangle. What is the area of hexagon DEF GHI ?

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 2

USA

AMC 10 2014

E A D

F

G

B

C

I

H

√ 12 + 3 3 (A) 4

9 (B) 2

(C) 3 +

√

3

√ 6+3 3 (D) 2

(E) 6

14 The y -intercepts, P and Q, of two perpendicular lines intersecting at the point A(6, 8) have a sum of zero. What is the area of AP Q? (A) 45 (B) 48 (C) 54 (D) 60 (E) 72

15 David drives from his home to the airport to catch a ?ight. He drives 35 miles in the ?rst hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home? (A) 140 (B) 175 (C) 210 (D) 245 (E) 280

16 In rectangle ABCD, AB = 1, BC = 2, and points E , F , and G are midpoints of BC , CD, and AD, respectively. Point H is the midpoint of GE . What is the area of the shaded region?

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 3

USA

AMC 10 2014

A

B

1

G

H

E

1

D

1 2

F

1 2

C

1 (A) 12

3 (B) 18

√

√ 2 (C) 12

√ (D)

3 12

(E)

1 6

17 Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? 13 7 5 2 1 (A) (B) (C) (D) (E) 6 72 36 24 9 18 A square in the coordinate plane has vertices whose y -coordinates are 0, 1, 4, and 5. What is the area of the square? (A) 16 (B) 17 (C) 25 (D) 26 (E) 27

19 Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XY contained in the cube with edge length 3? √ √ √ √ 3 33 2 33 (A) (B) 2 3 (C) (D) 4 (E) 3 2 5 3

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 4

USA

AMC 10 2014

X 1

2

3

4

Y

20 The product (8)(888 . . . 8), where the second factor has k digits, is an integer whose digits have a sum of 1000. What is k ? (A) 901 (B) 911 (C) 919 (D) 991 (E) 999

21 Positive integers a and b are such that the graphs of y = ax + 5 and y = 3x + b intersect the x-axis at the same point. What is the sum of all possible x-coordinates of these points of intersection? (A) ?20 (B) ?18 (C) ?15 (D) ?12 (E) ?8

22 In rectangle ABCD, AB = 20 and BC = 10. Let E be a point on CD such that ∠CBE = 15? . What is AE ? √ √ √ 20 3 (A) (B) 10 3 (C) 18 (D) 11 3 (E) 20 3 √ 23 A rectangular piece of paper whose length is 3 times the width has area A. The paper is divided into three sections along the opposite lengths, and then a dotted line is drawn from

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 5

USA

AMC 10 2014

the ?rst divider to the second divider on the opposite side as shown. The paper is then folded ?at along this dotted line to create a new shape with area B . What is the ratio B : A?

(A) 1 : 2

(B) 3 : 5

(C) 2 : 3

(D) 3 : 4

(E) 4 : 5

24 A sequence of natural numbers is constructed by listing the ?rst 4, then skipping one, listing the next 5, skipping 2, listing 6, skipping 3, and, on the nth iteration, listing n + 3 and skipping n. The sequence begins 1, 2, 3, 4, 6, 7, 8, 9, 10, 13. What is the 500, 000th number in the sequence? (A) 996, 506 (B) 996507 (C) 996508 (D) 996509 (E) 996510

25 The number 5867 is between 22013 and 22014 . How many pairs of integers (m, n) are there such that 1 ≤ m ≤ 2012 and 5n < 2m < 2m+2 < 5n+1 ? (A) 278 (B) 279 (C) 280 (D) 281 (E) 282

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 6

USA

AMC 10 2014

B

1 1. Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah’s coins worth? (A) 33 2 2. What is (A) 16 (B) 35

23 +23 ? 2?3 +2?3

(C) 37

(D) 39

(E) 41

(B) 24

(C) 32

(D) 48

(E) 64

3 Randy drove the ?rst third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-?fth on a dirt road. In miles, how long was Randy’s trip? (A) 30 (B)

400 11

(C)

75 2

(D) 40

(E)

300 7

4 Susie pays for 4 mu?ns and 3 bananas. Calvin spends twice as much paying for 2 mu?ns and 5 16 bananas. A mu?n is how many times as expensive as a banana? (A) 3 (B) 3 (C) 7 2 4 5 Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length o the square window?

(D) 2

(A) 26

(B) 28

(C) 30

(D) 32

(E) 34

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 7

USA

AMC 10 2014

6 Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 1 3 o? the regular price. What is the greatest number of balloons Orvin could buy? (A) 33 (B) 34 (C) 36 (D) 38 (E) 39

7 Suppose A > B > 0 and A is x% greater than B . What is x? (A) 100

A?B B

(B) 100

A+B B

(C) 100

A+B A

(D) 100

A?B A

(E) 100

A B

8 A truck travels b6 feet ever t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes? (A)

b 1080t

(B)

30t b

(C)

30b t

(D)

1 w 1 w

10t b

(E)

10b t

9 For real numbers w and z , + ?

1 z 1 z

= 2014.

What is

w +z w ?z

? (B)

?1 2014

(A) ? 2014

(C)

1 2014

(D) 1

(E) 2014

10 In the addition shown below A, B , C , and D are distinct digits. How many di?erent values are possible for D? ABBCB BCADA DBDDD (E) 9

+ (A) 2 (B) 4 (C) 7 (D) 8

11 For the consumer, a single discount of n% is more advantageous than any of the following discounts: (1) two successive 15% discounts (2) three successive 10% discounts (3) a 25% discount followed by a 5% discount What is the smallest possible positive integer value of n? (A) 27 (B) 28 (C) 29 (D) 31 (E) 33

12 The largest divisor of 2, 014, 000, 000 is itself. What is its ?fth largest divisor? (A) 125, 875, 000 (B) 201, 400, 000 (C) 251, 750, 000 (D) 402, 800, 000 (E) 503, 500, 000

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 8

USA

AMC 10 2014

13 Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of ABC ?

B

A

C

√ (A) 2 3

√ (B) 3 3

√ (C) 1 + 3 2

√ (D) 2 + 2 3

√ (E) 3 + 2 3

14 Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles were displayed on the odometer, where abc is a 3-digit number with a ≥ 1 and a + b + c ≤ 7. At the end of the trip, where the odometer showed cba miles. What is a2 + b2 + c2 ? (A) 26 (B) 27 (C) 36 (D) 37 (E) 41

15 In rectangle ABCD, DC = 2CB and points E and F lie on AB so that ED and F D trisect ∠ADC as shown. What is the ratio of the area of DEF to the area of rectangle ABCD?

A

E

F B

D

√ √ √ 3 3 16 √

C

(A)

3 6

(B)

6 8

(C)

(D)

1 3

(E)

2 4

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 9

USA

AMC 10 2014

16 Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? (A)

1 36

(B)

7 72

(C)

1 9

(D)

5 36

(E)

1 6

17 What is the greatest power of 2 that is a factor of 101002 ? 4501 ? (A) 21002 (B) 21003 (C) 21004 (D) 21005 (E) 21006

18 A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list? (A) 24 (B) 30 (C) 31 (D) 33 (E) 35

19 Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? (A)

1 6

(B)

1 4

(C)

√ 2? 2 2

(D)

1 3

(E)

1 2

20 For how many integers is the number x4 ? 51x2 + 50 negative? (A) 8 (B) 10 (C) 12 (D) 14 (E) 16

21 Trapezoid ABCD has parallel sides are of lengths 10 and 14. shorter diagonal of ABCD? √ (A) 10 6 (B) 25 (C)

sides AB or length 33 and CD of length 21. The other two The angles at A and B are acute. What is the length of the √ 8 10 √ (D) 18 2 (E) 26

22 Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 10

USA

AMC 10 2014

√ 1+ 2 (A) 4

√ (B)

5?1 2

√ (C)

3+1 4

√ 2 3 (D) 5

√ (E)

5 3

23 A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

3 (A) 2

√ 1+ 5 (B) 2

√ (C) 3

(D) 2

√ 3+ 5 (E) 2

24 The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can ?nd a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that di?er only by a rotation or a re?ection are considered the same. How many di?erent bad arrangements are there? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

25 In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on N pad 1. When the frog is on pad N , 0 < N < 10, it will jump to pad N ? 1 with probability 10 N and to pad N + 1 with probability 1 ? 10 . Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. what is the probability that the frog will escape being eaten by the snake? (A)

32 79

(B)

161 384

(C)

63 146

(D)

7 16

(E)

1 2

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 11

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