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Difference between revisions of "1984 AHSME Problems"

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{{AHSME Problems
 +
|year = 1984
 +
}}
 
==Problem 1==
 
==Problem 1==
  
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If <math> x, y </math>, and <math> y-\frac{1}{x} </math> are not <math> 0 </math>, then
 
If <math> x, y </math>, and <math> y-\frac{1}{x} </math> are not <math> 0 </math>, then
  
−
<math> \frac{x-\frac{1}{y}}{y-\frac{1}{x}} </math> equals
+
<cmath> \frac{x-\frac{1}{y}}{y-\frac{1}{x}} </cmath>  
 +
equals
  
 
<math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }\frac{x}{y} \qquad \mathrm{(C) \ } \frac{y}{x}\qquad \mathrm{(D) \ }\frac{x}{y}-\frac{y}{x} \qquad \mathrm{(E) \ } xy-\frac{1}{xy} </math>
 
<math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }\frac{x}{y} \qquad \mathrm{(C) \ } \frac{y}{x}\qquad \mathrm{(D) \ }\frac{x}{y}-\frac{y}{x} \qquad \mathrm{(E) \ } xy-\frac{1}{xy} </math>
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If the [[sequence]] <math> \{a_n\} </math> is defined by
 
If the [[sequence]] <math> \{a_n\} </math> is defined by
  
−
<math> a_1=2 </math>
+
<cmath>a_1=2,</cmath>
−
 
+
<cmath>a_{n+1}=a_n+2n \qquad (n \geq 1),</cmath>
−
<math> a_{n+1}=a_n+2n </math>
 
−
 
 
−
where <math> n\geq1 </math>.
 
  
−
Then <math> a_{100} </math> equals
+
then <math> a_{100} </math> equals
  
 
<math> \mathrm{(A) \ }9900 \qquad \mathrm{(B) \ }9902 \qquad \mathrm{(C) \ } 9904 \qquad \mathrm{(D) \ }10100 \qquad \mathrm{(E) \ } 10102 </math>
 
<math> \mathrm{(A) \ }9900 \qquad \mathrm{(B) \ }9902 \qquad \mathrm{(C) \ } 9904 \qquad \mathrm{(D) \ }10100 \qquad \mathrm{(E) \ } 10102 </math>
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==Problem 17==
 
==Problem 17==
−
A [[right triangle]] <math> ABC </math> with [[hypotenuse]] <math> AB </math> has side <math> AC=15 </math>. [[Altitude]] <math> CH </math> divides <math> AB </math> into segments <math> AH </math> and <math> HB </math>, with <math> HB=16 </math>. The [[area]] of <math> \triangle ABC </math> is:
+
A [[right triangle]] <math> ABC </math> with [[hypotenuse]] <math> AB </math> has side <math> AC=15 </math>. [[Altitude]] <math> CH </math> divides <math> AB </math> into segments <math> AH </math> and <math> HB </math>, with <math> HB=16 </math>. The [[area]] of <math> \triangle ABC </math> is
 +
 
 +
[[File:AHSME-1984-Q17.jpg]]
  
 
<math> \mathrm{(A) \ }120 \qquad \mathrm{(B) \ }144 \qquad \mathrm{(C) \ } 150 \qquad \mathrm{(D) \ }216 \qquad \mathrm{(E) \ } 144\sqrt{5} </math>
 
<math> \mathrm{(A) \ }120 \qquad \mathrm{(B) \ }144 \qquad \mathrm{(C) \ } 150 \qquad \mathrm{(D) \ }216 \qquad \mathrm{(E) \ } 144\sqrt{5} </math>
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==Problem 18==
 
==Problem 18==
−
A point <math> (x, y) </math> is to be chosen in the [[coordinate plane]] so that it is equally distant from the [[x-axis]], the [[y-axis]], and the [[line]] <math> x+y=2 </math>. Then <math> x </math> is
+
A point <math> (x, y) </math> is to be chosen in the [[coordinate plane]] so that it is equally distant from the [[x-axis|<math>x</math>-axis]], the [[y-axis|<math>y</math>-axis]], and the [[line]] <math> x+y=2 </math>. Then <math> x </math> is
  
−
<math> \mathrm{(A) \ }\sqrt{2}-1 \qquad \mathrm{(B) \ }\frac{1}{2} \qquad \mathrm{(C) \ } 2-\sqrt{2} \qquad \mathrm{(D) \ }1 \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math>
+
<math> \mathrm{(A) \ }\sqrt{2}-1 \qquad \mathrm{(B) \ }\frac{1}{2} \qquad \mathrm{(C) \ } 2-\sqrt{2} \qquad \mathrm{(D) \ }1 \qquad \mathrm{(E) \ } \text{not uniquely determined} </math>
  
 
[[1984 AHSME Problems/Problem 18|Solution]]
 
[[1984 AHSME Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
−
A box contains <math> 11 </math> balls, numbered <math> 1, 2, 3, ... 11 </math>. If <math> 6 </math> balls are drawn simultaneously at random, what is the [[probability]] that the sum of the numbers on the balls drawn is odd?
+
A box contains <math> 11 </math> balls, numbered <math> 1, 2, 3,\ldots,11 </math>. If <math> 6 </math> balls are drawn simultaneously at random, what is the [[probability]] that the sum of the numbers on the balls drawn is odd?
  
 
<math> \mathrm{(A) \ }\frac{100}{231} \qquad \mathrm{(B) \ }\frac{115}{231} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\frac{118}{231} \qquad \mathrm{(E) \ } \frac{6}{11} </math>
 
<math> \mathrm{(A) \ }\frac{100}{231} \qquad \mathrm{(B) \ }\frac{115}{231} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\frac{118}{231} \qquad \mathrm{(E) \ } \frac{6}{11} </math>
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==Problem 20==
 
==Problem 20==
−
The number of the distinct solutions to the equation
+
The number of the distinct solutions of the [[equation]] <math> |{x-|2x+1|}|=3 </math> is
−
 
 
−
<math> |x-|2x+1||=3 </math> is
 
  
 
<math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4 </math>
 
<math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4 </math>
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The number of triples <math> (a, b, c) </math> of [[Natural numbers|positive integers]] which satisfy the simultaneous [[Equation|equations]]
 
The number of triples <math> (a, b, c) </math> of [[Natural numbers|positive integers]] which satisfy the simultaneous [[Equation|equations]]
  
−
<math> ab+bc=44 </math>
+
<cmath> ab+bc=44,</cmath>
−
 
+
<cmath> ac+bc=23,</cmath>
−
<math> ac+bc=23 </math>  
 
  
 
is
 
is
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==Problem 22==
 
==Problem 22==
−
Let <math> a </math> and <math> c </math> be fixed [[Natural numbers|positive numbers]]. For each [[real number]] <math> t </math> let <math> (x_t, y_t) </math> be the [[vertex]] of the [[parabola]] <math> y=ax^2+bx+c </math>. If the set of the vertices <math> (x_t, y_t) </math> for all real numbers of <math> t </math> is graphed on the [[Cartesian plane|plane]], the [[graph]] is
+
Let <math> a </math> and <math> c </math> be fixed [[Natural numbers|positive numbers]]. For each [[real number]] <math> t </math> let <math> (x_t, y_t) </math> be the [[vertex]] of the [[parabola]] <math> y=ax^2+bx+c </math>. If the set of the vertices <math> (x_t, y_t) </math> for all real values of <math> t </math> is graphed on the [[Cartesian plane|plane]], the [[graph]] is
  
−
<math> \mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a parabola} \qquad \mathrm{(D) \ } \text{one branch of a hyperbola} \qquad </math> <math> \mathrm{(E) \ } \text{None of these} </math>
+
<math> \mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a parabola} \qquad \mathrm{(D) \ } \text{one branch of a hyperbola} \qquad \mathrm{(E) \ } \text{none of these} </math>
  
 
[[1984 AHSME Problems/Problem 22|Solution]]
 
[[1984 AHSME Problems/Problem 22|Solution]]
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==Problem 25==
 
==Problem 25==
−
The total [[area]] of all the [[faces]] of a [[Rectangular prism|rectangular solid]] is <math> 22\text{cm}^2 </math>, and the total length of all its [[edges]] is <math> 24\text{cm} </math>. Then the length in cm of any one of its [[Diagonal#Polyhedra|interior diagonals]] is
+
The total [[area]] of all the [[faces]] of a [[Rectangular prism|rectangular solid]] is <math> 22 \ \text{cm}^2 </math>, and the total length of all its [[edges]] is <math> 24 \ \text{cm} </math>. Then the length in cm of any one of its [[Diagonal#Polyhedra|interior diagonals]] is
  
−
<math> \mathrm{(A) \ }\sqrt{11} \qquad \mathrm{(B) \ }\sqrt{12} \qquad \mathrm{(C) \ } \sqrt{13}\qquad \mathrm{(D) \ }\sqrt{14} \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math>
+
<math> \mathrm{(A) \ }\sqrt{11} \qquad \mathrm{(B) \ }\sqrt{12} \qquad \mathrm{(C) \ } \sqrt{13}\qquad \mathrm{(D) \ }\sqrt{14} \qquad \mathrm{(E) \ } \text{not uniquely determined} </math>
  
 
[[1984 AHSME Problems/Problem 25|Solution]]
 
[[1984 AHSME Problems/Problem 25|Solution]]
  
 
==Problem 26==
 
==Problem 26==
−
In the [[obtuse triangle]] <math> ABC </math> with <math> \angle C>90^\circ </math>, <math> AM=MB </math>, <math> MD\perp BC </math>, and <math> EC\perp BC </math> (<math> D </math> is on <math> BC </math>, <math> E </math> is on <math> AB </math>, and <math> M </math> is on <math> EB </math>). If the [[area]] of <math> \triangle ABC </math> is <math> 24 </math>, then the area of <math> \triangle BED </math> is
+
In the [[obtuse triangle]] <math> ABC </math>, <math> AM=MB </math>, <math> MD\perp BC </math>, <math> EC\perp BC </math>. If the [[area]] of <math> \triangle ABC </math> is <math> 24 </math>, then the area of <math> \triangle BED </math> is
  
−
<math> \mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math>
+
[[File:AHSME-1984-Q26.jpg]]
 +
 
 +
<math> \mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{not uniquely determined} </math>
  
 
[[1984 AHSME Problems/Problem 26|Solution]]
 
[[1984 AHSME Problems/Problem 26|Solution]]
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==Problem 27==
 
==Problem 27==
 
In <math> \triangle ABC </math>, <math> D </math> is on <math> AC </math> and <math> F </math> is on <math> BC </math>. Also, <math> AB\perp AC </math>, <math> AF\perp BC </math>, and <math> BD=DC=FC=1 </math>. Find <math> AC </math>.
 
In <math> \triangle ABC </math>, <math> D </math> is on <math> AC </math> and <math> F </math> is on <math> BC </math>. Also, <math> AB\perp AC </math>, <math> AF\perp BC </math>, and <math> BD=DC=FC=1 </math>. Find <math> AC </math>.
 +
 +
[[File:AHSME-1984-Q27.jpg]]
  
 
<math> \mathrm{(A) \ }\sqrt{2} \qquad \mathrm{(B) \ }\sqrt{3} \qquad \mathrm{(C) \ } \sqrt[3]{2} \qquad \mathrm{(D) \ }\sqrt[3]{3} \qquad \mathrm{(E) \ } \sqrt[4]{3} </math>
 
<math> \mathrm{(A) \ }\sqrt{2} \qquad \mathrm{(B) \ }\sqrt{3} \qquad \mathrm{(C) \ } \sqrt[3]{2} \qquad \mathrm{(D) \ }\sqrt[3]{3} \qquad \mathrm{(E) \ } \sqrt[4]{3} </math>
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==Problem 30==
 
==Problem 30==
 
For any [[complex number]] <math> w=a+bi </math>, <math> |w| </math> is defined to be the [[real number]] <math> \sqrt{a^2+b^2} </math>. If <math> w=\cos40^\circ+i\sin40^\circ </math>, then
 
For any [[complex number]] <math> w=a+bi </math>, <math> |w| </math> is defined to be the [[real number]] <math> \sqrt{a^2+b^2} </math>. If <math> w=\cos40^\circ+i\sin40^\circ </math>, then
 +
<math> |w+2w^2+3w^3+...+9w^9|^{-1} </math> equals
  
−
<math> |w+2w^2+3w^3+...+9w^9|^{-1} </math>
+
<math> \mathrm{(A) \ }\frac{1}{9}\sin40^\circ \qquad \mathrm{(B) \ }\frac{2}{9}\sin20^\circ \qquad \mathrm{(C) \ } \frac{1}{9}\cos40^\circ \qquad \mathrm{(D) \ }\frac{1}{18}\cos20^\circ \qquad \mathrm{(E) \ } \text{none of these} </math>
−
 
 
−
equals
 
−
 
 
−
<math> \mathrm{(A) \ }\frac{1}{9}\sin40^\circ \qquad \mathrm{(B) \ }\frac{2}{9}\sin20^\circ \qquad \mathrm{(C) \ } \frac{1}{9}\cos40^\circ \qquad \mathrm{(D) \ }\frac{1}{18}\cos20^\circ \qquad \mathrm{(E) \ } \text{None of these} </math>
 
  
 
[[1984 AHSME Problems/Problem 30|Solution]]
 
[[1984 AHSME Problems/Problem 30|Solution]]

Latest revision as of 14:00, 19 February 2020

1984 AHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have 90 minutes working time to complete the test.
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

Problem 1

$\frac{1000^2}{252^2-248^2}$ equals

$\mathrm{(A) \ }62500 \qquad \mathrm{(B) \ }1000 \qquad \mathrm{(C) \ } 500\qquad \mathrm{(D) \ }250 \qquad \mathrm{(E) \ } \frac{1}{2}$

Solution

Problem 2

If $x, y$, and $y-\frac{1}{x}$ are not $0$, then

\[\frac{x-\frac{1}{y}}{y-\frac{1}{x}}\] equals

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }\frac{x}{y} \qquad \mathrm{(C) \ } \frac{y}{x}\qquad \mathrm{(D) \ }\frac{x}{y}-\frac{y}{x} \qquad \mathrm{(E) \ } xy-\frac{1}{xy}$

Solution

Problem 3

Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then

$\mathrm{(A) \ }100<n\leq110 \qquad \mathrm{(B) \ }110<n\leq120 \qquad \mathrm{(C) \ } 120<n\leq130 \qquad \mathrm{(D) \ }130<n\leq140 \qquad \mathrm{(E) \ } 140<n\leq150$

Solution

Problem 4

A rectangle intersects a circle as shown: $AB = 4$, $BC = 5$ and $DE = 3$. Then $EF$ equals AHSME-1984-Q4.jpg

$\mathrm{(A) \ }6 \qquad \mathrm{(B) \ }7 \qquad \mathrm{(C) \ }\frac{20}{3} \qquad \mathrm{(D) \ }8 \qquad \mathrm{(E) \ }9$

Solution

Problem 5

The largest integer $n$ for which $n^{200}<5^{300}$ is

$\mathrm{(A) \ }8 \qquad \mathrm{(B) \ }9 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }11 \qquad \mathrm{(E) \ } 12$

Solution

Problem 6

In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls and teachers, respectively, then the total number of boys, girls and teachers can be represented by the expression

$\mathrm{(A) \ }31b \qquad \mathrm{(B) \ }\frac{37}{27}b \qquad \mathrm{(C) \ } 13g \qquad \mathrm{(D) \ }\frac{37}{27}g \qquad \mathrm{(E) \ } \frac{37}{27}t$

Solution

Problem 7

When Dave walks to school, he averages $90$ steps per minute, each of his steps $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?

$\mathrm{(A) \ }14 \frac{2}{9} \text{ minutes} \qquad \mathrm{(B) \ }15 \text{ minutes}\qquad \mathrm{(C) \ } 18 \text{ minutes}\qquad \mathrm{(D) \ }20 \text{ minutes} \qquad \mathrm{(E) \ } 22 \frac{2}{9} \text{ minutes}$

Solution

Problem 8

Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$ and $\angle CDA=60^\circ$. The length of $DC$ is

AHSME-1984-Q8.jpg

$\mathrm{(A) \ }7+\frac{2}{3}\sqrt{3} \qquad \mathrm{(B) \ }8 \qquad \mathrm{(C) \ } 9 \frac{1}{2} \qquad \mathrm{(D) \ }8+\sqrt{3} \qquad \mathrm{(E) \ } 8+3\sqrt{3}$

Solution

Problem 9

The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is

$\mathrm{(A) \ }31 \qquad \mathrm{(B) \ }30 \qquad \mathrm{(C) \ } 29 \qquad \mathrm{(D) \ }28 \qquad \mathrm{(E) \ } 27$

Solution

Problem 10

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i, {-2}+i$ and ${-1}-2i$. The fourth number is

$\mathrm{(A) \ }2+i \qquad \mathrm{(B) \ }2-i \qquad \mathrm{(C) \ } 1-2i \qquad \mathrm{(D) \ }{-1}+2i \qquad \mathrm{(E) \ } {-2}-i$

Solution

Problem 11

A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result when one starts with an entry $x\not=0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then $y$ equals

$\mathrm{(A) \ }x^{((-2)^n)} \qquad \mathrm{(B) \ }x^{2n} \qquad \mathrm{(C) \ } x^{-2n} \qquad \mathrm{(D) \ }x^{-(2^n)} \qquad \mathrm{(E) \ } x^{((-1)^n2n)}$

Solution

Problem 12

If the sequence $\{a_n\}$ is defined by

\[a_1=2,\] \[a_{n+1}=a_n+2n \qquad (n \geq 1),\]

then $a_{100}$ equals

$\mathrm{(A) \ }9900 \qquad \mathrm{(B) \ }9902 \qquad \mathrm{(C) \ } 9904 \qquad \mathrm{(D) \ }10100 \qquad \mathrm{(E) \ } 10102$

Solution

Problem 13

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$ equals

$\mathrm{(A) \ }\sqrt{2}+\sqrt{3}-\sqrt{5} \qquad \mathrm{(B) \ }4-\sqrt{2}-\sqrt{3} \qquad \mathrm{(C) \ } \sqrt{2}+\sqrt{3}+\sqrt{6}-5 \qquad$

$\mathrm{(D) \ }\frac{1}{2}(\sqrt{2}+\sqrt{5}-\sqrt{3}) \qquad \mathrm{(E) \ } \frac{1}{3}(\sqrt{3}+\sqrt{5}-\sqrt{2})$

Solution

Problem 14

The product of all real roots of the equation $x^{\log_{10}{x}}=10$ is

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }-1 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }10^{-1} \qquad \mathrm{(E) \ } \text{None of these}$

Solution

Problem 15

If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is

$\mathrm{(A) \ }18^\circ \qquad \mathrm{(B) \ }30^\circ \qquad \mathrm{(C) \ } 36^\circ \qquad \mathrm{(D) \ }45^\circ \qquad \mathrm{(E) \ } 60^\circ$

Solution

Problem 16

The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 8$

Solution

Problem 17

A right triangle $ABC$ with hypotenuse $AB$ has side $AC=15$. Altitude $CH$ divides $AB$ into segments $AH$ and $HB$, with $HB=16$. The area of $\triangle ABC$ is

AHSME-1984-Q17.jpg

$\mathrm{(A) \ }120 \qquad \mathrm{(B) \ }144 \qquad \mathrm{(C) \ } 150 \qquad \mathrm{(D) \ }216 \qquad \mathrm{(E) \ } 144\sqrt{5}$

Solution

Problem 18

A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the $x$-axis, the $y$-axis, and the line $x+y=2$. Then $x$ is

$\mathrm{(A) \ }\sqrt{2}-1 \qquad \mathrm{(B) \ }\frac{1}{2} \qquad \mathrm{(C) \ } 2-\sqrt{2} \qquad \mathrm{(D) \ }1 \qquad \mathrm{(E) \ } \text{not uniquely determined}$

Solution

Problem 19

A box contains $11$ balls, numbered $1, 2, 3,\ldots,11$. If $6$ balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?

$\mathrm{(A) \ }\frac{100}{231} \qquad \mathrm{(B) \ }\frac{115}{231} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\frac{118}{231} \qquad \mathrm{(E) \ } \frac{6}{11}$

Solution

Problem 20

The number of the distinct solutions of the equation $|{x-|2x+1|}|=3$ is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4$

Solution

Problem 21

The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations

\[ab+bc=44,\] \[ac+bc=23,\]

is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4$

Solution

Problem 22

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real values of $t$ is graphed on the plane, the graph is

$\mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a parabola} \qquad \mathrm{(D) \ } \text{one branch of a hyperbola} \qquad \mathrm{(E) \ } \text{none of these}$

Solution

Problem 23

$\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}$ equals

$\mathrm{(A) \ }\tan{10^\circ}+\tan{20^\circ} \qquad \mathrm{(B) \ }\tan{30^\circ} \qquad \mathrm{(C) \ } \frac{1}{2}(\tan{10^\circ}+\tan{20^\circ}) \qquad \mathrm{(D) \ }\tan{15^\circ} \qquad \mathrm{(E) \ } \frac{1}{4}\tan{60^\circ}$

Solution

Problem 24

If $a$ and $b$ are positive real numbers and each of the equations $x^2+ax+2b=0$ and $x^2+2bx+a=0$ has real roots, then the smallest possible value of $a+b$ is

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ } 6$

Solution

Problem 25

The total area of all the faces of a rectangular solid is $22 \ \text{cm}^2$, and the total length of all its edges is $24 \ \text{cm}$. Then the length in cm of any one of its interior diagonals is

$\mathrm{(A) \ }\sqrt{11} \qquad \mathrm{(B) \ }\sqrt{12} \qquad \mathrm{(C) \ } \sqrt{13}\qquad \mathrm{(D) \ }\sqrt{14} \qquad \mathrm{(E) \ } \text{not uniquely determined}$

Solution

Problem 26

In the obtuse triangle $ABC$, $AM=MB$, $MD\perp BC$, $EC\perp BC$. If the area of $\triangle ABC$ is $24$, then the area of $\triangle BED$ is

AHSME-1984-Q26.jpg

$\mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{not uniquely determined}$

Solution

Problem 27

In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB\perp AC$, $AF\perp BC$, and $BD=DC=FC=1$. Find $AC$.

AHSME-1984-Q27.jpg

$\mathrm{(A) \ }\sqrt{2} \qquad \mathrm{(B) \ }\sqrt{3} \qquad \mathrm{(C) \ } \sqrt[3]{2} \qquad \mathrm{(D) \ }\sqrt[3]{3} \qquad \mathrm{(E) \ } \sqrt[4]{3}$

Solution

Problem 28

The number of distinct pairs of integers $(x, y)$ such that $0<x<y$ and $\sqrt{1984}=\sqrt{x}+\sqrt{y}$ is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ }4 \qquad \mathrm{(E) \ } 7$

Solution

Problem 29

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x, y)$ which satisfy $(x-3)^2+(y-3)^2=6$.

$\mathrm{(A) \ }3+2\sqrt{2} \qquad \mathrm{(B) \ }2+\sqrt{3} \qquad \mathrm{(C) \ } 3\sqrt{3} \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 6+2\sqrt{3}$

Solution

Problem 30

For any complex number $w=a+bi$, $|w|$ is defined to be the real number $\sqrt{a^2+b^2}$. If $w=\cos40^\circ+i\sin40^\circ$, then $|w+2w^2+3w^3+...+9w^9|^{-1}$ equals

$\mathrm{(A) \ }\frac{1}{9}\sin40^\circ \qquad \mathrm{(B) \ }\frac{2}{9}\sin20^\circ \qquad \mathrm{(C) \ } \frac{1}{9}\cos40^\circ \qquad \mathrm{(D) \ }\frac{1}{18}\cos20^\circ \qquad \mathrm{(E) \ } \text{none of these}$

Solution


See also

1984 AHSME (Problems • Answer Key • Resources)
Preceded by
1983 AHSME 		Followed by
1985 AHSME
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All AHSME Problems and Solutions


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