AoPS Community 1955 AMC 12/AHSME

AMC 12/AHSME 1955

www.artofproblemsolving.com/community/c4819
by AIME15, rrusczyk

1 Which one of the following is not equivalent to 0.000000375?

(A) 3.75 × 10−7 (B) 3 34 × 10−7 (C) 375 × 10−9
3
(D) 8 × 10−7 3
(E) 80000000

2 The smaller angle between the hands of a clock at 12 : 25 p.m. is:

(A) 132◦ 300 (B) 137◦ 300 (C) 150◦ (D) 137◦ 320 (E) 137◦

3 If each number in a set of ten numbers is increased by 20, the arithmetic mean (average) of
the original ten numbers:
(A) remains the same (B) is increased by 20 (C) is increased by 200
(D) is increased by 10 (E) is increased by 2

4 The equality x−1

1 2
= x−2 is satisfied by:
(A) no real values of x (B) either x = 1 or x = 2 (C) only x = 1
(D) only x = 2 (E) only x = 0

5 y varies inversely as the square of x. When y = 16, x = 1. When x = 8, y equals:

(A) 2 (B) 128 (C) 64 (D) 14 (E) 1024

6 A merchant buys a number of oranges at 3 for 10 cents and an equal number at 5 for 20 cents.
To ”break even” he must sell all at:
(A) 8 for 30 cents (B) 3 for 11 cents (C) 5 for 18 cents
(D) 11 for 40 cents (E) 13 for 50 cents

7 If a worker receives a 20 percent cut in wages, he may regain his original pay exactly by ob-
taining a raise of:
(A) 20 percent (B) 25 percent (C) 22 21 percent (D) $20 (E) $25

8 The graph of x2 − 4y 2 = 0:
(A) is a hyperbola intersecting only the x -axis
(B) is a hyperbola intersecting only the y -axis
(C) is a hyperbola intersecting neither axis
(D) is a pair of straight lines
(E) does not exist

9 A circle is inscribed in a triangle with sides 8, 15, and 17. The radius of the circle is:
(A) 6 (B) 2 (C) 5 (D) 3 (E) 7

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AoPS Community 1955 AMC 12/AHSME

10 How many hours does it take a train traveling at an average rate of 40 mph between stops to
travel a miles it makes n stops of m minutes each?
(A) 3a+2mn
120 (B) 3a + 2mn (C) 3a+2mn
12 (D) a+mn
40 (E) a+40mn
40

11 The negation of the statement ”No slow learners attend this school” is:
(A) All slow learners attend this school
(B) All slow learners do not attend this school
(C) Some slow learners attend this school
(D) Some slow learners do not attend this school
(E) No slow learners do not attend this school
√ √
12 The solution of 5x − 1 + x − 1 = 2 is:
(A) x = 2, x = 1 (B) x = 23 (C) x = 2 (D) x = 1 (E) x = 0

−4 −4
13 The fraction aa−2 −b
−b
−2 is equal to:
−6
(A) a − b −6 (B) a−2 − b−2 (C) a−2 + b−2
(D) a2 + b2 (E) a2 − b2

14 The length of rectangle R is 10 percent more than the side of square S. The width of the rect-
angle is 10 percent less than the side of the square. The ratio of the areas, R:S, is:
(A) 99 : 100 (B) 101 : 100 (C) 1 : 1 (D) 199 : 200 (E) 201 : 200

15 The ratio of the areas of two concentric circles is 1 : 3. If the radius of the smaller is r, then
the difference between the radii is best approximated by:
(A) 0.41r (B) 0.73 (C) 0.75 (D) 0.73r (E) 0.75r

16 The value of a+b3

when a = 4 and b = −4 is:
(A) 3 (B) 38 (C) 0 (D) any finite number (E) meaningless

17 If log x − 5 log 3 = −2, then x equals:

(A) 1.25 (B) 0.81 (C) 2.43 (D) 0.8 (E) either 0.8 or 1.25
√
18 The discriminant of the equation x2 + 2x 3 + 3 = 0 is zero. Hence, its roots are:
(A) real and equal (B) rational and equal (C) rational and unequal
(D) irrational and unequal (E) imaginary

19 Two numbers whose sum is 6 and the absolute value of whose difference is 8 are roots of the
equation:
(A) x2 − 6x + 7 = 0 (B) x2 − 6x − 7 = 0 (C) x2 + 6x − 8 = 0
2
(D) x − 6x + 8 = 0 2
(E) x + 6x − 7 = 0

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AoPS Community 1955 AMC 12/AHSME
√
20 The expression 25 − t2 + 5 equals zero for:
(A) no real or imaginary values of t (B) no real values of t only
(C) no imaginary values of t only (D) t = 0 (E) t = ±5

21 Represent the hypotenuse of a right triangle by c and the area by A. The atltidue on the hy-
potenuse is:
2
(A) Ac (B) 2A
c
A
(C) 2c (D) Ac (E) cA2

22 On a $10000 order a merchant has a choice between three successive discounts of 20%, 20%,
and 10% and three successive discounts of 40%, 5%, and 5%. By choosing the better offer, he
can save:
(A) nothing at all (B) $440 (C) $330 (D) $345 (E) $360

23 In checking the petty cash a clerk counts q quarters, d dimes, n nickels, and c cents. Later he
discovers that x of the nickels were counted as quarters and x of the dimes were counted as
cents. To correct the total obtained the clerk must:
(A) make no correction (B) subtract 11 cents (C) subtract 11x cents
(D) add 11x cents (E) add x cents

24 The function 4x2 − 12x − 1:

(A) always increases as x increases
(B) always decreases as x decreases to 1
(C) cannot equal 0
(D) has a maximum value when x is negative
(E) has a minimum value of -10

25 One of the factors of x4 + 2x2 + 9 is:

(A) x2 + 3 (B) x + 1 (C) x2 − 3 (D) x2 − 2x − 3 (E) none of these

26 Mr. A owns a house worth $10000. He sells it to Mr. B at 10% profit. Mr. B sells the house back
to Mr. A at a 10% loss. Then:
(A) Mr. A comes out even (B) Mr. A makes $100 (C) Mr. A makes $1000
(D) Mr. B loses $100 (E) none of the above is correct

27 If r and s are the roots of x2 − px + q = 0, then r2 + s2 equals:

(A) p2 + 2q (B) p2 − 2q (C) p2 + q 2 (D) p2 − q 2 (E) p2

28 On the same set of axes are drawn the graph of y = ax2 + bx + c and the graph of the equation
obtained by replacing x by −x in the given equation. If b 6= 0 and c 6= 0 these two graphs
intersect:
(A) in two points, one on the x-axis and one on the y-axis
(B) in one point located on neither axis

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AoPS Community 1955 AMC 12/AHSME

(C) only at the origin

(D) in one point on the x-axis
(E) in one point on the y-axis

29 In the figure, P A is tangent to semicircle SAR; P B is tangent to semicircle RBT ; SRT is a

straight line; the arcs are indicated in the figure. Angle AP B is measured by:

A
c
P B
a
b d

S R T

(A) 12 (a − b) (B) 12 (a + b) (C) (c − a) − (d − b) (D) a − b (E) a + b

√ √
30 Each of the equations 3x2 − 2 = 25, (2x − 1)2 = (x − 1)2 , x2 − 7 = x − 1 has:
(A) two integral roots (B) no root greater than 3 (C) no root zero
(D) only one root (E) one negative root and one positive root

31 An equilateral triangle whose side is 2 is divided into a triangle and a trapezoid by a line drawn
parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original
triangle, the length of the median of the trapezoid is: √ √
√
6
√ √ √
2+ 2
(A) 2 (B) 2 (C) 2 + 2 (D) 2 (E) 2 3−
2
6

32 If the discriminant of ax2 + 2bx + c = 0 is zero, then another true statement about a, b, and c is
that:
(A) they form an arithmetic progression
(B) they form a geometric progression
(C) they are unequal
(D) they are all negative numbers
(E) only b is negative and a and c are positive

33 Henry starts a trip when the hands of the clock are together between 8 a.m. and 9 a.m. He
arrives at his destination between 2 p.m. and 3 p.m. when the hands of the clock are exactly
180◦ apart. The trip takes:
(A) 6 hr. (B) 6 hr. 43-7/11 min. (C) 5 hr. 16-4/11 min. (D) 6 hr. 30 min. (E) none of these

34 A 6-inch and 18-inch diameter pole are placed together and bound together with wire. The

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AoPS Community 1955 AMC 12/AHSME

length√of the shortest wire√that will go around

√ them is:
(A) 12 3 + 16π (B) 12 3 + 7π (C) 12 3 + 14π
(D) 12 + 15π (E) 24π

35 Three boys agree to divide a bag of marbles in the following manner. The first boy takes one
more than half the marbles. The second takes a third of the number remaining. The third boy
finds that he is left with twice as many marbles as the second boy. The original number of
marbles:
(A) is none of the following (B) cannot be determined from the given data
(C) is 20 or 26 (D) is 14 or 32 (E) is 8 or 38

36 A cylindrical oil tank, lying horizontally, has an interior length of 10 feet and an interior diameter
of 6 feet. If the rectangular surface of the oil has an area of 40 square feet, the depth of the oil
is: √ √ √ √
(A) 5 (B)√2 5 (C)
√ 3 − 5 (D) 3 + 5
(E) either 3 − 5 or 3 + 5

37 A three-digit number has, from left to right, the digits h, t, and u, with h > u. When the number
with the digits reversed is subtracted from the original number, the units’ digit in the difference
of r. The next two digits, from right to left, are:
(A) 5 and 9 (B) 9 and 5 (C) impossible to tell (D) 5 and 4 (E) 4 and 5

38 Four positive integers are given. Select any three of these integers, find their arithmetic aver-
age, and add this result to the fourth integer. Thus the numbers 29, 23, 21, and 17 are obtained.
One of the original integers is:
(A) 19 (B) 21 (C) 23 (D) 29 (E) 17

39 If y = x2 + px + q, then if the least possible value of y is zero q is equal to:

2 2
(A) 0 (B) p4 (C) p2 (D) − p2 (E) p4 − q

40 The fractions ax+b

cx+d and d are unequal if:
b

(A) a = c = 1, x 6= 0 (B) a = b = 0 (C) a = c = 0

(D) x = 0 (E) ad = bc

41 A train traveling from Aytown to Beetown meets with an accident after 1 hr. It is stopped for
2 hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown 2 hr. late. If the
1

train had covered 80 miles more before the accident, it would have been just 1 hr. late. The
usual rate of the train is:
(A) 20 mph (B) 30 mph (C) 40 mph (D) 50 mph (E) 60 mph
q q
42 If a, b, and c are positive integers, the radicals a + cb and a cb are equal when and only when:

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AoPS Community 1955 AMC 12/AHSME
b(a2 −1)
(A) a = b = c = 1 (B) a = b and c = a = 1 (C) c = 2
(D) a = b and c is any value (E) a = b and c = a − 1

43 The pairs of values of x and y that are the common solutions of the equations y = (x + 1)2
and xy + y = 1 are:
(A) 3 real pairs (B) 4 real pairs (C) 4 imaginary pairs
(D) 2 real and 2 imaginary pairs (E) 1 real and 2 imaginary pairs

44 In circle O chord AB is produced so that BC equals a radius of the circle. CO is drawn and
extended to D. AO is drawn. Which of the following expresses the relationship between x and
y?

A B C
y

x
O
D

(A) x = 3y
(B) x = 2y
(C) x = 60◦
(D) there is no special relationship between x and y
(E) x = 2y or x = 3y, depending upon the length of AB

45 Given a geometric sequence with the first term 6= 0 and r 6= 0 and an arithmetic sequence with
the first term = 0. A third sequence 1, 1, 2 . . . is formed by adding corresponding terms of the
two given sequences. The sum of the first ten terms of the third sequence is:
(A) 978 (B) 557 (C) 467 (D) 1068
(E) not possible to determine from the information given

46 The graphs of 2x + 3y − 6 = 0, 4x − 3y − 6 = 0, x = 2, and y = 2

3 intersect in:
(A) 6 points (B) 1 point (C) 2 points (D) no points
(E) an unlimited number of points

47 The expressions a + bc and (a + b)(a + c) are:

(A) always equal (B) never equal (C) equal whenever a + b + c = 1
(D) equal when a + b + c = 0 (E) equal only when a = b = c = 0

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48 Given triangle ABC with medians AE, BF , CD; F H parallel and equal to AE; BH and HE are
drawn; F E extended meets BH in G. Which one of the following statements is not necessarily
correct?
(A) AEHF is a parallelogram (B) HE = HG
(C) BH = DC 3
(D) F G = 4 AB (E) F G is a median of triangle BF H

2
49 The graphs of y = xx−2
−4
and y = 2x intersect in:
(A) 1 point whose abscissa is 2 (B) 1 point whose abscissa is 0
(C) no points (D) two distinct points (E) two identical points

50 In order to pass B going 40 mph on a two-lane highway, A, going 50 mph, must gain 30 feet.
Meantime, C, 210 feet from A, is headed toward him at 50 mph. If B and C maintain their
speeds, then, in order to pass safely, A must increase his speed by:
(A) 30 mph (B) 10 mph (C) 5 mph (D) 15 mph (E) 3 mph

–
These problems are copyright © Mathematical Association of America (http://maa.org).

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