AoPS Community 1964 AMC 12/AHSME

AMC 12/AHSME 1964

www.artofproblemsolving.com/community/c4828
by Mrdavid445, TheMaskedMagician, JSGandora, rrusczyk

1 What is the value of [log10 (5 log10 100)]2 ?

(A) log10 50 (B) 25 (C) 10 (D) 2 (E) 1

2 The graph of x2 − 4y 2 = 0 is:

(A) a parabola (B) an ellipse (C) a pair of straight lines (D) a point (E) none of these

3 When a positive integer x is divided by a positive integer y, the quotient is u and the remainder
is v, where u and v are integers. What is the remainder when x + 2uy is divided by y?
(A) 0 (B) 2u (C) 3u (D) v (E) 2v

4 The expression

P +Q P −Q
−
P −Q P +Q

where P = x + y and Q = x − y, is equivalent to:

x2 −y 2 x2 −y 2 x2 +y 2 x2 +y 2
(A) xy (B) 2xy (C) 1 (D) xy (E) 2xy

5 If y varies directly as x, and if y = 8 when x = 4, the value of y when x = −8 is:

(A) − 16 (B) − 4 (C) − 2 (D) 4k, k = ±1, ±2, . . . (E) 16k, k = ±1, ±2, . . .

6 If x, 2x + 2, 3x + 3, . . . are in geometric progression, the fourth term is:

(A) − 27 (B) − 13 21 (C) 12 (D) 13 21 (E) 27

7 Let n be the number of real values of p for which the roots of

x2 − px + p = 0

are equal. Then n equals:

(A) 0 (B) 1 (C) 2 (D) a finite number greater than 2 (E) an infinitely large number

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The smaller root of the equation x − 3 3 3 1

= 0 is:





8 4 x− 4 + x− 4 x− 2
3 1 5 3
(A) − 4 (B) 2 (C) 8 (D) 4 (E) 1

9 A jobber buys an article at $24 less 12 12 %. He then wishes to sell the article at a gain of 33 13 %
of his cost after allowing a 20% discount on his marked price. At what price, in dollars, should
the article be marked?
(A) 25.20 (B) 30.00 (C) 33.60 (D) 40.00 (E) none of these

10 Given a square side of length s. On a diagonal as base a triangle with three unequal sides is
constructed so that its area equals that of the square. The length of the altitude drawn to the
base is:
√ √ √ √
(A) s 2 (B) s/ 2 (C) 2s (D) 2 s (E) 2/ s

11 Given 2x = 8y+1 and 9y = 3x−9 , find the value of x + y.

(A) 18 (B) 21 (C) 24 (D) 27 (E) 30

12 Which of the following is the negation of the statement: For all x of a certain set, x2 > 0?
(A) For all x, x2 < 0 (B) For all x, x2 ≤ 0 (C) For no x, x2 > 0 (D) For some x, x2 > 0
(E) For some x, x ≤ 0
2

13 A circle is inscribed in a triangle with side lengths 8, 13, and 17. Let the segments of the side
of length 8, made by a point of tangency, be r and s, with r < s. What is the ratio r : s?
(A) 1 : 3 (B) 2 : 5 (C) 1 : 2 (D) 2 : 3 (E) 3 : 4

14 A farmer bought 749 sheeps. He sold 700 of them for the price paid for the 749 sheep. The
remaining 49 sheep were sold at the same price per head as the other 700. Based on the cost,
the percent gain on the entire transaction is:
(A) 6.5 (B) 6.75 (C) 7 (D) 7.5 (E) 8

15 A line through the point (−a, 0) cuts from the second quadrant a triangular region with area T .
The equation of the line is:
(A) 2T x+a2 y+2aT = 0 (B) 2T x−a2 y+2aT = 0 (C) 2T x + a2 y − 2aT = 0 (D) 2T x − a2 y − 2aT =

16 Let f (x) = x2 + 3x + 2 and let S be the set of integers {0, 1, 2, . . . , 25}. The number of members
s of S such that f (s) has remainder zero when divided by 6 is:
(A) 25 (B) 22 (C) 21 (D) 18 (E) 17

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17 Given the distinct points P (x1 , y1 ), Q(x2 , y2 ) and R(x1 + x2 , y1 + y2 ). Line segments are drawn
connecting these points to each other and to the origin 0. Of the three possibilities: (1) parallel-
ogram (2) straight line (3) trapezoid, figure OP RQ, depending upon the location of the points
P, Q, and R, can be:
(A) (1) only (B) (2) only (C) (3) only (D) (1) or (2) only (E) all three

18 Let n be the number of pairs of values of b and c such that 3x + by + c = 0 and cx − 2y + 12 = 0

have the same graph. Then n is:
(A) 0 (B) 1 (C) 2 (D) finite but more than 2 (E) greater than any finite number

x2 +3xy
19 If 2x − 3y − z = 0 and x + 3y − 14z = 0, z 6= 0, the numerical value of y 2 +z 2
is:

(A) 7 (B) 2 (C) 0 (D) − 20/17 (E) − 2

20 The sum of the numerical coefficients of all the terms in the expansion of (x − 2y)18 is:
(A) 0 (B) 1 (C) 19 (D) − 1 (E) − 19

21 If logb2 x + logx2 b = 1, b > 0, b 6= 1, x 6= 1, then x equals:

√
(A) 1/b2 (B) 1/b (C) b2 (D) b (E) b

22 Given parallelogram ABCD with E the midpoint of diagonal BD. Point E is connected to a
point F in DA so that DF = 31 DA. What is the ratio of the area of triangle DF E to the area of
quadrilateral ABEF ?
(A) 1 : 2 (B) 1 : 3 (C) 1 : 5 (D) 1 : 6 (E) 1 : 7

23 Two numbers are such that their difference, their sum, and their product are to one another as
1 : 7 : 24. The product of the two numbers is:
(A) 6 (B) 12 (C) 24 (D) 48 (E) 96

24 Let y = (x − a)2 + (x − b)2 , a, b constants. For what value of x is y a minimum?

√ q
a2 +b2
(A) a+b
2 (B) a + b (C) ab (D) 2 (E) a+b
2ab

25 The set of values of m for which x2 +3xy+x+my−m has two factors, with integer coefficients,
which are linear in x and y, is precisely:
(A) 0, 12, −12 (B) 0, 12 (C) 12, −12 (D) 12 (E) 0

26 In a ten-mile race First beats Second by 2 miles and First beats Third by 4 miles. If the runners
maintain constant speeds throughout the race, by how many miles does Second beat Third?

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(A) 2 (B) 2 14 (C) 2 21 (D) 2 34 (E) 3

27 If x is a real number and |x − 4| + |x − 3| < a where a > 0, then:

(A) 0 < a < .01 (B) .01 < a < 1 (C) 0 < a < 1
(D) 0 < a ≤ 1 (E) a > 1

28 The sum of n terms of an arithmetic progression is 153, and the common difference is 2. If the
first interm is an integer, and n > 1, then the number of possible values for n is:
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

1
29 In this figure ∠RF S = ∠F DR, F D = 4 inches, DR = 6 inches, F R = 5 inches, F S = 7
2
inches. The length of RS, in inches, is:

R
6
D
5
4
F 7 12 S

1 1
(A) undetermined (B) 4 (C) 5 (D) 6 (E) 6
2 4
√ √
30 If (7 + 4 3)x2 + (2 + 3)x − 2 = 0, the larger root minus the smaller root is:
√ √ √ √ √
(A) − 2 + 3 3 (B) 2 − 3 (C) 6 + 3 3 (D) 6 − 3 3 (E) 3 3 + 2

31 Let √ √ !n √ √ !n
5+3 5 1+ 5 5−3 5 1− 5
f (n) = + .
10 2 10 2

Then f (n + 1) − f (n − 1), expressed in terms of f (n), equals:

1 1
(A) f (n) (B) f (n) (C) 2f (n) + 1 (D) f 2 (n) (E) (f 2 (n) − 1)
2 2

a+b c+d
32 If = , then:
b+c d+a
(A) a must equal c (B) a + b + c + d must equal zero
(C) either a = c or a + b + c + d = 0, or both

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(D) a + b + c + d 6= 0 if a = c (E) a(b + c + d) = c(a + b + d)

33 P is a point interior to rectangle ABCD and such that P A = 3 inches, P D = 4 inches, and
P C = 5 inches. Then P B, in inches, equals:
√ √ √ √
(A) 2 3 (B) 3 2 (C) 3 3 (D) 4 2 (E) 2

D C

4 5

3 P

A B

√
34 If n is a multiple of 4, the sum s = 1 + 2i + 3i2 + ... + (n + 1)in , where i = −1, equals:
(A) 1 + i (B) 12 (n + 2) (C) 12 (n + 2 − ni)
(D) 12 [(n + 1)(1 − i) + 2] (E) 18 (n2 + 8 − 4ni)

35 The sides of a triangle are of lengths 13, 14, and 15. The altitudes of the triangle meet at point
H. If AD is the altitude to the side length 14, what is the ratio HD : HA?
(A) 3 : 11 (B) 5 : 11 (C) 1 : 2 (D) 2 : 3 (E) 25 : 33

36 In this figure the radius of the circle is equal to the altitude of the equilateral triangle ABC.
The circle is made to roll along the side AB, remaining tangent to it at a variable point T and
intersecting lines AC and BC in variable points M and N , respectively. Let n be the number
of degrees in arc M T N . Then n, for all permissible positions of the circle:
(A) varies from 30◦ to 90◦
(B) varies from 30◦ to 60◦
(C) varies from 60◦ to 90◦
(D) remains constant at 30◦
(E) remains constant at 60◦

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M N
A T B

37 Given two positive number a, b such that a < b. Let A.M. be their arithmetic mean and let G.M.
be their positive geometric mean. Then A.M. minus G.M. is always less than:
(b + a)2 (b + a)2 (b − a)2
(A) (B) (C)
ab 8b ab
(b − a)2 (b − a)2
(D) (E)
8a 8b

38 The sides P Q and P R of triangle P QR are respectively of lengths 4 inches, and 7 inches. The
median P M is 3 21 inches. Then QR, in inches, is:
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

39 The magnitudes of the sides of triangle ABC are a, b, and c, as shown, with c ≤ b ≤ a. Through
interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A0 , B 0 ,
C 0 , respectively. Let s = AA0 + BB 0 + CC 0 . Then, for all positions of point P , s is less than:
(A) 2a + b (B) 2a + c (C) 2b + c (D) a + 2b (E) a + b + c

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AoPS Community 1964 AMC 12/AHSME
B

a
A′
c C′

B′
A b C

40 A watch loses 2 12 minutes per day. It is set right at 1 P.M. on March 15. Let n be the positive
correction, in minutes, to be added to the time shown by the watch at a given time. When the
watch shows 9 A.M. on March 21, n equals:
(A) 14 14
23
1
(B) 14 14 (C) 13 101
115
83
(D) 13 115 (E) 13 13
23

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