Difference between revisions of "1983 AHSME Problems"

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label("$P$", P, S);
 
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label("$40^\circ$", C+(-0.15,0), NW);
 
label("$40^\circ$", C+(-0.15,0), NW);
label("$10^\circ$", B+(0,0.05), dir(200);
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Revision as of 09:57, 9 July 2018

Problem 1

If $x \neq 0, \frac x{2} = y^2$ and $\frac{x}{4} = 4y$, then $x$ equals

$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 32\qquad \textbf{(D)}\ 64\qquad \textbf{(E)}\ 128$

Solution

Problem 2

Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \text{cm}$ from P?

$\text{(A)} \ 1 \qquad  \text{(B)} \ 2 \qquad  \text{(C)} \ 3 \qquad  \text{(D)} \ 4 \qquad  \text{(E)} \ 8$

Solution

Problem 3

Three primes $p,q$, and $r$ satisfy $p+q = r$ and $1 < p < q$. Then $p$ equals

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 17$

Solution

Problem 4

Position $A,B,C,D,E,F$ such that $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length of $1$ and it is given that $\measuredangle FAB = \measuredangle BCD = 60^\circ$. The area of the figure is

$\text{(A)} \ \frac{\sqrt 3}{2} \qquad  \text{(B)} \ 1 \qquad  \text{(C)} \ \frac{3}{2} \qquad  \text{(D)}\ \sqrt{3}\qquad \text{(E)}$

Solution

Problem 5

Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then $\tan B$ is

$\textbf{(A)}\ \frac{3}{5}\qquad \textbf{(B)}\ \frac{\sqrt 5}{3}\qquad \textbf{(C)}\ \frac{2}{\sqrt 5}\qquad \text{(D)}\ \sqrt{3}\qquad \text{(E)}\ 2$

Solution

Problem 6

When $x^5, x+\frac{1}{x}$ and $1+\frac{2}{x} + \frac{3}{x^2}$ are multiplied, the product is a polynomial of degree.

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Solution

Problem 7

Alice sells an item at $ $10$ less than the list price and receives $10\%$ of her selling price as her commission. Bob sells the same item at $ $20$ less than the list price and receives $20\%$ of his selling price as his commission. If they both get the same commission, then the list price in dollars is

$\textbf{(A) } 20\qquad \textbf{(B) } 30\qquad \textbf{(C) } 50\qquad \textbf{(D) } 70\qquad \textbf{(E) } 100$


Solution

Problem 8

Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1, f(-x)$ is

$\textbf{(A)}\ \frac{1}{f(x)}\qquad \textbf{(B)}\ -f(x)\qquad \textbf{(C)}\ \frac{1}{f(-x)}\qquad \textbf{(D)}\ -f(-x)\qquad \textbf{(E)}\ f(x)$

Solution

Problem 9

In a certain population the ratio of the number of women to the number of men is $11$ to $10$. If the average (arithmetic mean) age of the women is $34$ and the average age of the men is $32$, then the average age of the population is

$\textbf{(A)}\ 32\frac{9}{10}\qquad \textbf{(B)}\ 32\frac{20}{21}\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 33\frac{1}{21}\qquad \textbf{(E)}\ 33\frac{1}{10}$

Solution

Problem 10

Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BD$ at points $D$ and $E$, respectively. The length of $AE$ is

$\text{(A)} \ \frac{3}{2} \qquad  \text{(B)} \ \frac{5}{3} \qquad  \text{(C)} \ \frac{\sqrt 3}{2} \qquad  \text{(D)}\ \sqrt{3}\qquad \text{(E)}\ \frac{2+\sqrt 3}{2}$

Solution

Problem 11

Simplify $\sin (x-y) \cos y + \cos (x-y) \sin y$.

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \sin x\qquad \textbf{(C)}\ \cos x\qquad \textbf{(D)}\ \sin x \cos 2y\qquad \textbf{(E)}\ \cos x\cos 2y$

Solution

Problem 12

If $\log_2 \Big(\log_3 (\log_2 x) \Big) = 0$, then $x^{-1/2}$ equals

$\text{(A)} \ \frac{1}{3} \qquad  \text{(B)} \ \frac{1}{2 \sqrt 3} \qquad  \text{(C)}\ \frac{1}{3\sqrt 3}\qquad \text{(D)}\ \frac{1}{\sqrt{42}}\qquad \text{(E)}\ \text{none of these}$

Solution

Problem 13

If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is zero, then $x^2+y^2+z^2$ equals:

$\textbf{(A)}\ \frac{ab+ac+bc}{abc}\qquad \textbf{(B)}\ \frac{a^2+b^2+c^2}{abc}\qquad \textbf{(C)}\ \frac{(a+b+c)^2}{abc}\qquad \textbf{(D)}\ \frac{(ab+ac+bc)^2}{abc}\qquad \textbf{(E)}\ \frac{(ab)^2+(ac)^2+(bc)^2}{abc}$

Solution

Problem 14

The units digit of $3^{1001}* 7^{1002}* 13^{1003}$ is

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Solution

Problem 15

Three balls marked $1,2$, and $3$, are placed in an urn. One ball is drawn, its number is recorded, then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times?

$\text{(A)} \ \frac{1}{27} \qquad  \text{(B)} \ \frac{1}{8} \qquad  \text{(C)} \ \frac{1}{7} \qquad  \text{(D)}\ \frac{1}{6}\qquad \text{(E)}\ \frac{1}{3}$

Solution

Problem 16

Let $x = .123456789101112....998999$, where the digits are obtained by writing the integers $1$ through $999$ in order. The $1983$rd digit to the right of the decimal point is

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Solution

Problem 17

The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of $F$. Which one?

$\text{(A)} \ A \qquad  \text{(B)} \ B \qquad  \text{(C)} \ C \qquad  \text{(D)} \ D \qquad  \text{(E)} \ E$

Solution

Problem 18

Let $f$ be a polynomial function such that, for all real $x$, $f(x^2 + 1) = x^4 + 5x^2 + 3$. For all real $x, f(x^2-1)$ is

$\textbf{(A)}\ x^4+5x^2+1\qquad \textbf{(B)}\ x^4+x^2-3\qquad \textbf{(C)}\ x^4-5x^2+1\qquad \textbf{(D)}\ x^4+x^2+3\qquad\\ \textbf{(E)}\ \text{None of these}$

Solution

Problem 19

Point $D$ is on side $CB$ of triangle $ABC$. If $\angle{CAD} = \angle{DAB} = 60^\circ\mbox{, }AC = 3\mbox{ and }AB = 6$, then the length of $AD$ is

$\text{(A)} \ 2 \qquad  \text{(B)} \ 2.5 \qquad  \text{(C)} \ 3 \qquad  \text{(D)} \ 3.5 \qquad  \text{(E)} \ 4$

Solution

Problem 20

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily

$\text{(A)} \ pq \qquad  \text{(B)} \ \frac{1}{pq} \qquad  \text{(C)} \ \frac{p}{q^2} \qquad  \text{(D)}\ \frac{q}{p^2}\qquad \text{(E)}\ \frac{p}{q}$

Solution

Problem 21

Find the smallest positive number from the numbers below

$\text{(A)} \ 10-3\sqrt{11} \qquad  \text{(B)} \ 3\sqrt{11}-10 \qquad  \text{(C)}\ 18-5\sqrt{13}\qquad\\ \text{(D)}\ 51-10\sqrt{26}\qquad \text{(E)}\ 10\sqrt{26}-51$

Solution

Problem 22

Consider the two functions $f(x) = x^2+2bx+1$ and $g(x) = 2a(x+b)$, where the variable $x$ and the constants $a$ and $b$ are real numbers. Each such pair of the constants a and b may be considered as a point $(a,b)$ in an $ab$-plane. Let S be the set of such points $(a,b)$ for which the graphs of $y = f(x)$ and $y = g(x)$ do NOT intersect (in the $xy$- plane.). The area of $S$ is

$\text{(A)} \ 1 \qquad  \text{(B)} \ \pi \qquad  \text{(C)} \ 4 \qquad  \text{(D)} \ 4 \pi \qquad  \text{(E)} \ \infty$

Solution

Problem 23

In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is

[asy] size(250);defaultpen(linewidth(0.7)); real alpha=5.797939254, x=71.191836; int i; for(i=0; i<5; i=i+1) { real r=8*(sqrt(6)/2)^i; draw(Circle((x+r)*dir(alpha), r)); x=x+2r; } real x=71.191836+40+20*sqrt(6), r=18; pair A=tangent(origin, (x+r)*dir(alpha), r, 1), B=tangent(origin, (x+r)*dir(alpha), r, 2); pair A1=300*dir(origin--A), B1=300*dir(origin--B); draw(B1--origin--A1); pair X=(69,-5), X1=reflect(origin, (x+r)*dir(alpha))*X, Y=(200,-5), Y1=reflect(origin, (x+r)*dir(alpha))*Y, Z=(130,0), Z1=reflect(origin, (x+r)*dir(alpha))*Z; clip(X--Y--Y1--X1--cycle); label("$L_2$", Z, S); label("$L_1$", Z1, dir(2*alpha)*dir(90));[/asy]

$\text{(A)} \ 12 \qquad  \text{(B)} \ 12.5 \qquad  \text{(C)} \ 13 \qquad  \text{(D)} \ 13.5 \qquad  \text{(E)} \ 14$

Solution

Problem 24

How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal?

$\text{(A)} \ \text{none} \qquad  \text{(B)} \ 1 \qquad  \text{(C)} \ 2 \qquad  \text{(D)} \ 4 \qquad  \text{(E)} \ \infty$

Solution

Problem 25

If $60^a = 3$ and $60^b = 5$, then $12^{[(1-a-b)/2(1-b)]}$

$\text{(A)} \ \sqrt{3} \qquad  \text{(B)} \ 2 \qquad  \text{(C)} \ \sqrt{5} \qquad  \text{(D)} \ 3 \qquad  \text{(E)} \ \sqrt{12}$

Solution

Problem 26

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval

$\textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad \textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad \textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad \textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad \textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$

Solution

Problem 27

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)

$\textbf{(A)}\ \frac{5}{2}\qquad \textbf{(B)}\ 9 - 4\sqrt{5}\qquad \textbf{(C)}\ 8\sqrt{10} - 23\qquad \textbf{(D)}\ 6-\sqrt{15}\qquad \textbf{(E)}\ 10\sqrt{5}-20$

Solution

Problem 28

Triangle $\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$, are on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$2$", (2,0), S); label("$3$", (7,0), S);[/asy]

$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ \frac{5}{3}\sqrt{10}\qquad \textbf{(E)}\ \text{not uniquely determined}$

Solution

Problem 29

A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?

$\textbf{(A)}\ 1 + \sqrt{2}\qquad \textbf{(B)}\ 2\sqrt{2}\qquad \textbf{(C)}\ 2 + \sqrt{2}\qquad \textbf{(D)}\ 3\sqrt{2}\qquad \textbf{(E)}\ 3+\sqrt{2}$

Solution

Problem 30

Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw(M--N^^C--A--P--B--C^^Arc(origin,1,0,180)); markscalefactor=0.03; draw(anglemark(C,A,P)); draw(anglemark(C,B,P)); pair point=C; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, S); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$P$", P, S); label("$40^\circ$", C+(-0.15,0), NW); label("$10^\circ$", B+(0,0.05), dir(200)); label("$10^\circ$", A+(0.05,0.02), dir(330));[/asy]

$\textbf{(A)}\ 10^{\circ}\qquad \textbf{(B)}\ 15^{\circ}\qquad \textbf{(C)}\ 20^{\circ}\qquad \textbf{(D)}\ 25^{\circ}\qquad \textbf{(E)}\ 30^{\circ}$

See also

1983 AHSME (ProblemsAnswer KeyResources)
Preceded by
1982 AHSME
Followed by
1984 AHSME
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
All AHSME Problems and Solutions


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