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Difference between revisions of "1979 AHSME Problems"

(Created page with "== Problem 1 == <asy> draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle); draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle); label("$F$",(0,0),E); label("$A$",(-2,1),W); label("$B$",(2,1),E)...")
 
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 +
{{AHSME Problems
 +
|year = 1979
 +
}}
 
== Problem 1 ==
 
== Problem 1 ==
  
 
<asy>
 
<asy>
−
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
 
−
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
 
−
label("$F$",(0,0),E);
 
−
label("$A$",(-2,1),W);
 
−
label("$B$",(2,1),E);
 
−
label("$C$", (2,-1),E);
 
−
label("$D$",(-2,-1),WSW);
 
−
label("$E$",(-2,0),W);
 
−
label("$G$",(0,-1),S);
 
−
//Credit to TheMaskedMagician for the diagram</asy><asy>
 
 
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
 
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
 
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
 
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
Line 202: Line 195:
 
== Problem 13 ==
 
== Problem 13 ==
 
    
 
    
−
13  The inequality <math>y-x<\sqrt{x^2}</math> is satisfied if and only if
+
The inequality <math>y-x<\sqrt{x^2}</math> is satisfied if and only if
  
−
<math>\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad
+
<math>\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(C) }y^2<2xy\qquad \textbf{(D) }y<0\qquad \textbf{(E) }x>0\text{ and }y<2x </math>     
−
\textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad
 
−
\textbf{(C) }y^2<2xy\qquad
 
−
\textbf{(D) }y<0\qquad
 
−
\textbf{(E) }x>0\text{ and }y<2x </math>     
 
  
 
[[1979 AHSME Problems/Problem 13|Solution]]
 
[[1979 AHSME Problems/Problem 13|Solution]]
−
+
 
 
== Problem 14 ==
 
== Problem 14 ==
 
    
 
    
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label("$D$",D,S);
 
label("$D$",D,S);
 
label("x",X,fontsize(5pt));
 
label("x",X,fontsize(5pt));
−
//Credit to TheMaskedMagician for the diagram</asy>
+
//Credit to TheMaskedMagician for the diagram
 +
</asy>
  
 
Points <math>A , B, C</math>, and <math>D</math> are distinct and lie, in the given order, on a straight line.  
 
Points <math>A , B, C</math>, and <math>D</math> are distinct and lie, in the given order, on a straight line.  
Line 279: Line 269:
 
<math>\textbf{I. }x<\frac{z}{2}\qquad\\
 
<math>\textbf{I. }x<\frac{z}{2}\qquad\\
 
\textbf{II. }y<x+\frac{z}{2}\qquad\\
 
\textbf{II. }y<x+\frac{z}{2}\qquad\\
−
\textbf{III. }y<\frac{z}{2}</math>\\
+
\textbf{III. }y<\frac{z}{2}\qquad </math>
 +
 
  
 
<math>\textbf{(A) }\textbf{I. }\text{only}\qquad
 
<math>\textbf{(A) }\textbf{I. }\text{only}\qquad
Line 306: Line 297:
 
Find the sum of the squares of all real numbers satisfying the equation <math>x^{256}-256^{32}=0</math>.
 
Find the sum of the squares of all real numbers satisfying the equation <math>x^{256}-256^{32}=0</math>.
  
−
\textbf{(A) }8\qquad
+
<math>\textbf{(A) }8\qquad
 
\textbf{(B) }128\qquad
 
\textbf{(B) }128\qquad
 
\textbf{(C) }512\qquad
 
\textbf{(C) }512\qquad
 
\textbf{(D) }65,536\qquad
 
\textbf{(D) }65,536\qquad
−
\textbf{(E) }2(256^{32})    
+
\textbf{(E) }2(256^{32})</math>
 
   
 
   
 
[[1979 AHSME Problems/Problem 19|Solution]]
 
[[1979 AHSME Problems/Problem 19|Solution]]
−
+
 
 
== Problem 20 ==
 
== Problem 20 ==
 
   
 
   
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Find the least possible distance between a pair of points <math>P</math> and <math>Q</math>, where <math>P</math> is on edge <math>AB</math> and <math>Q</math> is on edge <math>CD</math>.
 
Find the least possible distance between a pair of points <math>P</math> and <math>Q</math>, where <math>P</math> is on edge <math>AB</math> and <math>Q</math> is on edge <math>CD</math>.
  
−
<math>\textbf{(A) }\frac{1}{2}\qquad
 
−
\textbf{(B) }\frac{3}{4}\qquad
 
−
\textbf{(C) }\frac{\sqrt{2}}{2}\qquad
 
−
\textbf{(D) }\frac{\sqrt{3}}{2}\qquad
 
−
\textbf{(E) }\frac{\sqrt{3}}{3}</math>
 
  
 
<asy>
 
<asy>
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//Credit to TheMaskedMagician for the diagram</asy>
 
//Credit to TheMaskedMagician for the diagram</asy>
  
 +
 +
<math>\textbf{(A) }\frac{1}{2}\qquad
 +
\textbf{(B) }\frac{3}{4}\qquad
 +
\textbf{(C) }\frac{\sqrt{2}}{2}\qquad
 +
\textbf{(D) }\frac{\sqrt{3}}{2}\qquad
 +
\textbf{(E) }\frac{\sqrt{3}}{3}</math>
  
 
[[1979 AHSME Problems/Problem 23|Solution]]
 
[[1979 AHSME Problems/Problem 23|Solution]]
Line 403: Line 395:
 
\textbf{(C) }24.6\qquad
 
\textbf{(C) }24.6\qquad
 
\textbf{(D) }24.8\qquad
 
\textbf{(D) }24.8\qquad
−
\textbf{(E) }25</math>\\
+
\textbf{(E) }25</math>
  
 
    
 
    
Line 477: Line 469:
 
<math>\textbf{(A) }3r-2\qquad
 
<math>\textbf{(A) }3r-2\qquad
 
\textbf{(B) }r^2\qquad
 
\textbf{(B) }r^2\qquad
−
\textbf{(C) }r+\sqrt{3(r-1)}\qquad
+
\textbf{(C) }r+\sqrt{3(r-1)}\qquad\\
 
\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad
 
\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad
 
\textbf{(E) }\text{none of these}  </math>   
 
\textbf{(E) }\text{none of these}  </math>   
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For each positive number <math>x</math>, let  
 
For each positive number <math>x</math>, let  
−
<math>f(x)=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}</math>.  
+
<math>f(x)=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}
 +
{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}</math>.  
 
The minimum value of <math>f(x)</math> is
 
The minimum value of <math>f(x)</math> is
  
−
\textbf{(A) }1\qquad
+
<math>\textbf{(A) }1\qquad
 
\textbf{(B) }2\qquad
 
\textbf{(B) }2\qquad
 
\textbf{(C) }3\qquad
 
\textbf{(C) }3\qquad
 
\textbf{(D) }4\qquad
 
\textbf{(D) }4\qquad
−
\textbf{(E) }6     
+
\textbf{(E) }6    </math>
 
    
 
    
 
[[1979 AHSME Problems/Problem 29|Solution]]
 
[[1979 AHSME Problems/Problem 29|Solution]]
Line 534: Line 527:
  
 
== See also ==
 
== See also ==
−
* [[AHSME]]
+
 
−
* [[AHSME Problems and Solutions]]
 
 
* [[AMC 12 Problems and Solutions]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
{{AHSME box|year=1979|before=[[1978 AHSME]]|after=[[1980 AHSME]]}} 
 +
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:34, 19 March 2022

1979 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

[asy] draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle); draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle); label("$F$",(0,0),E); label("$A$",(-2,1),W); label("$B$",(2,1),E); label("$C$", (2,-1),E); label("$D$",(-2,-1),WSW); label("$E$",(-2,0),W); label("$G$",(0,-1),S); //Credit to TheMaskedMagician for the diagram [/asy]

If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is

$\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }12\qquad \textbf{(D) }18\qquad \textbf{(E) }24$

Solution

Problem 2

For all non-zero real numbers $x$ and $y$ such that $x-y=xy, \frac{1}{x}-\frac{1}{y}$ equals

$\textbf{(A) }\frac{1}{xy}\qquad \textbf{(B) }\frac{1}{x-y}\qquad \textbf{(C) }0\qquad \textbf{(D) }-1\qquad \textbf{(E) }y-x$

Solution

Problem 3

[asy] real s=sqrt(3)/2; draw(box((0,0),(1,1))); draw((1+s,0.5)--(1,1)); draw((1+s,0.5)--(1,0)); draw((0,1)--(1+s,0.5)); label("$A$",(1,1),N); label("$B$",(1,0),S); label("$C$",(0,0),W); label("$D$",(0,1),W); label("$E$",(1+s,0.5),E); //Credit to TheMaskedMagician for the diagram[/asy]

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?

$\textbf{(A) }10\qquad \textbf{(B) }12.5\qquad \textbf{(C) }15\qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Solution

Problem 4

For all real numbers $x, x[x\{x(2-x)-4\}+10]+1=$

$\textbf{(A) }-x^4+2x^3+4x^2+10x+1\qquad \textbf{(B) }-x^4-2x^3+4x^2+10x+1\qquad\\ \textbf{(C) }-x^4-2x^3-4x^2+10x+1\qquad \textbf{(D) }-x^4-2x^3-4x^2-10x+1\qquad\\ \textbf{(E) }-x^4+2x^3-4x^2+10x+1$

Solution

Problem 5

Find the sum of the digits of the largest even three digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.

$\textbf{(A) }22\qquad \textbf{(B) }23\qquad \textbf{(C) }24\qquad \textbf{(D) }25\qquad \textbf{(E) }26$

Solution

Problem 6

$\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7=$

$\textbf{(A) }-\frac{1}{64}\qquad \textbf{(B) }-\frac{1}{16}\qquad \textbf{(C) }0\qquad \textbf{(D) }\frac{1}{16}\qquad \textbf{(E) }\frac{1}{64}$

Solution

Problem 7

The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is

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

Solution

Problem 8

Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.

$\textbf{(A) }\frac{\pi}{4}\qquad \textbf{(B) }\frac{3\pi}{4}\qquad \textbf{(C) }\pi\qquad \textbf{(D) }\frac{3\pi}{2}\qquad \textbf{(E) }2\pi$

Solution

Problem 9

The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals

$\textbf{(A) }\sqrt[7]{12}\qquad \textbf{(B) }2\sqrt[7]{12}\qquad \textbf{(C) }\sqrt[7]{32}\qquad \textbf{(D) }\sqrt[12]{32}\qquad \textbf{(E) }2\sqrt[12]{32}$

Solution

Problem 10

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is

$\textbf{(A) }6\qquad \textbf{(B) }2\sqrt{6}\qquad \textbf{(C) }\frac{8\sqrt{3}}{3}\qquad \textbf{(D) }3\sqrt{3}\qquad \textbf{(E) }4\sqrt{3}$

Solution

Problem 11

Find a positive integral solution to the equation $\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}$

$\textbf{(A) }110\qquad \textbf{(B) }115\qquad \textbf{(C) }116\qquad \textbf{(D) }231\qquad\\ \textbf{(E) }\text{The equation has no positive integral solutions.}$

Solution

Problem 12

[asy] size(200); pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0); draw(arc(O,1, 0, 180)); EE=midpoint(arc(O,1, 0, 90)); draw(A--EE); draw(A--D); B=intersectionpoint(arc(O,1, 180, 0),EE--A); draw(O--EE); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,S);label("$D$",D,E);label("$E$",EE,NE);label("$O$",O,S);label("$45^\circ$",(0.25,0.1),fontsize(10pt)); //Credit to TheMaskedMagician for the diagram[/asy]

In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the measure of $\measuredangle BAO$ is

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

Solution

Problem 13

The inequality $y-x<\sqrt{x^2}$ is satisfied if and only if

$\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(C) }y^2<2xy\qquad \textbf{(D) }y<0\qquad \textbf{(E) }x>0\text{ and }y<2x$

Solution

Problem 14

In a certain sequence of numbers, the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. The sum of the third and the fifth numbers in the sequence is

$\textbf{(A) }\frac{25}{9}\qquad \textbf{(B) }\frac{31}{15}\qquad \textbf{(C) }\frac{61}{16}\qquad \textbf{(D) }\frac{576}{225}\qquad \textbf{(E) }34$

Solution

Problem 15

Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is

$\textbf{(A) }\frac{p+q}{2}\qquad \textbf{(B) }\frac{p^2+q^2}{p+q}\qquad \textbf{(C) }\frac{2pq}{p+q}\qquad \textbf{(D) }\frac{2(p^2+pq+q^2)}{3(p+q)}\qquad \textbf{(E) }\frac{p+q+2pq}{p+q+2}$

Solution

Problem 16

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is

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

Solution

Problem 17

[asy] size(200); dotfactor=3; pair A=(0,0),B=(1,0),C=(2,0),D=(3,0),X=(1.2,0.7); draw(A--D); dot(A);dot(B);dot(C);dot(D); draw(arc((0.4,0.4),0.4,180,110),arrow = Arrow(TeXHead)); draw(arc((2.6,0.4),0.4,0,70),arrow = Arrow(TeXHead)); draw(B--X,dotted); draw(C--X,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("x",X,fontsize(5pt)); //Credit to TheMaskedMagician for the diagram [/asy]

Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$, respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?

$\textbf{I. }x<\frac{z}{2}\qquad\\ \textbf{II. }y<x+\frac{z}{2}\qquad\\ \textbf{III. }y<\frac{z}{2}\qquad$


$\textbf{(A) }\textbf{I. }\text{only}\qquad \textbf{(B) }\textbf{II. }\text{only}\qquad \textbf{(C) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad \textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$

Solution

Problem 18

To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$?

$\textbf{(A) }\frac{8}{7}\qquad \textbf{(B) }\frac{9}{7}\qquad \textbf{(C) }\frac{10}{7}\qquad \textbf{(D) }\frac{11}{7}\qquad \textbf{(E) }\frac{12}{7}$

Solution

Problem 19

Find the sum of the squares of all real numbers satisfying the equation $x^{256}-256^{32}=0$.

$\textbf{(A) }8\qquad \textbf{(B) }128\qquad \textbf{(C) }512\qquad \textbf{(D) }65,536\qquad \textbf{(E) }2(256^{32})$

Solution

Problem 20

If $a=\tfrac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\arctan a + \arctan b$ equals

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

Solution

Problem 21

The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the circle to the area of the triangle is

$\textbf{(A) }\frac{\pi r}{h+2r}\qquad \textbf{(B) }\frac{\pi r}{h+r}\qquad \textbf{(C) }\frac{\pi}{2h+r}\qquad \textbf{(D) }\frac{\pi r^2}{r^2+h^2}\qquad \textbf{(E) }\text{none of these}$

Solution

Problem 22

Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$.

$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }3\qquad \textbf{(D) }9\qquad \textbf{(E) }\infty$

Solution

Problem 23

The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.


[asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern("hatch")); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); //Credit to TheMaskedMagician for the diagram[/asy]


$\textbf{(A) }\frac{1}{2}\qquad \textbf{(B) }\frac{3}{4}\qquad \textbf{(C) }\frac{\sqrt{2}}{2}\qquad \textbf{(D) }\frac{\sqrt{3}}{2}\qquad \textbf{(E) }\frac{\sqrt{3}}{3}$

Solution

Problem 24

Sides $AB,~ BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5$, and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5}$, then side $AD$ has length

  • A polygon is called “simple” if it is not self intersecting.

$\textbf{(A) }24\qquad \textbf{(B) }24.5\qquad \textbf{(C) }24.6\qquad \textbf{(D) }24.8\qquad \textbf{(E) }25$


Solution

Problem 25

If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals

$\textbf{(A) }\frac{1}{256}\qquad \textbf{(B) }-\frac{1}{16}\qquad \textbf{(C) }1\qquad \textbf{(D) }-16\qquad \textbf{(E) }256$

Solution

Problem 26

The function $f$ satisfies the functional equation $f(x) +f(y) = f(x + y ) - xy - 1$ for every pair $x,~ y$ of real numbers. If $f( 1) = 1$, then the number of integers $n \neq 1$ for which $f ( n ) = n$ is

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

Solution

Problem 27

An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will not have distinct positive real roots?

$\textbf{(A) }\frac{106}{121}\qquad \textbf{(B) }\frac{108}{121}\qquad \textbf{(C) }\frac{110}{121}\qquad \textbf{(D) }\frac{112}{121}\qquad \textbf{(E) }\text{none of these}$

Solution

Problem 28

[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP("$A$",A)); D(MP("$B$",B)); D(MP("$C$",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE); //Credit to TheMaskedMagician for the diagram[/asy]

Circles with centers $A ,~ B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals

$\textbf{(A) }3r-2\qquad \textbf{(B) }r^2\qquad \textbf{(C) }r+\sqrt{3(r-1)}\qquad\\ \textbf{(D) }1+\sqrt{3(r^2-1)}\qquad \textbf{(E) }\text{none of these}$

Solution

Problem 29

For each positive number $x$, let $f(x)=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2} {\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$. The minimum value of $f(x)$ is

$\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }4\qquad \textbf{(E) }6$

Solution

Problem 30

[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle); D(MP("$E$",EE,N)--MP("$D$",D,S)); D(D);D(EE); MA("80^\circ",8,D,EE,C,0.1); MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA("100^\circ",8,A,B,C,0.1,0); MA("60^\circ",8,C,A,B,0.1,0); //Credit to TheMaskedMagician for the diagram [/asy]

In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ, \measuredangle ABC = 100^\circ, \measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals

$\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad \textbf{(B) }\frac{\sqrt{3}}{8}\qquad \textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad \textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad \textbf{(E) }\frac{1}{8}$


Solution

See also

1979 AHSME (Problems • Answer Key • Resources)
Preceded by
1978 AHSME 		Followed by
1980 AHSME
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All AHSME Problems and Solutions


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