Difference between revisions of "1982 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1982 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
When the polynomial <math>x^3-2</math> is divided by the polynomial <math>x^2-2</math>, the remainder is | When the polynomial <math>x^3-2</math> is divided by the polynomial <math>x^2-2</math>, the remainder is | ||
− | <math>\ | + | <math>\textbf{(A)} \ 2 \qquad |
− | \ | + | \textbf{(B)} \ -2 \qquad |
− | \ | + | \textbf{(C)} \ -2x-2 \qquad |
− | \ | + | \textbf{(D)} \ 2x+2 \qquad |
− | \ | + | \textbf{(E)} \ 2x-2</math> |
[[1982 AHSME Problems/Problem 1|Solution]] | [[1982 AHSME Problems/Problem 1|Solution]] | ||
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If a number eight times as large as <math>x</math> is increased by two, then one fourth of the result equals | If a number eight times as large as <math>x</math> is increased by two, then one fourth of the result equals | ||
− | <math>\ | + | <math>\textbf{(A)} \ 2x + \frac{1}{2} \qquad |
− | \ | + | \textbf{(B)} \ x + \frac{1}{2} \qquad |
− | \ | + | \textbf{(C)} \ 2x+2 \qquad |
− | \ | + | \textbf{(D)}\ 2x+4 \qquad |
− | \ | + | \textbf{(E)}\ 2x+16 </math> |
[[1982 AHSME Problems/Problem 2|Solution]] | [[1982 AHSME Problems/Problem 2|Solution]] | ||
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Evaluate <math>(x^x)^{(x^x)}</math> at <math>x = 2</math>. | Evaluate <math>(x^x)^{(x^x)}</math> at <math>x = 2</math>. | ||
− | <math>\ | + | <math>\textbf{(A)} \ 16 \qquad |
− | \ | + | \textbf{(B)} \ 64 \qquad |
− | \ | + | \textbf{(C)} \ 256 \qquad |
− | \ | + | \textbf{(D)} \ 1024 \qquad |
− | \ | + | \textbf{(E)} \ 65,536 </math> |
[[1982 AHSME Problems/Problem 3|Solution]] | [[1982 AHSME Problems/Problem 3|Solution]] | ||
Line 40: | Line 43: | ||
measured in square centimeters. The radius of the semicircle, measured in centimeters, is | measured in square centimeters. The radius of the semicircle, measured in centimeters, is | ||
− | <math>\ | + | <math>\textbf{(A)} \ \pi \qquad |
− | \ | + | \textbf{(B)} \ \frac{2}{\pi} \qquad |
− | \ | + | \textbf{(C)} \ 1 \qquad |
− | \ | + | \textbf{(D)} \ \frac{1}{2}\qquad |
− | \ | + | \textbf{(E)} \ \frac{4}{\pi}+2 </math> |
− | |||
− | |||
[[1982 AHSME Problems/Problem 4|Solution]] | [[1982 AHSME Problems/Problem 4|Solution]] | ||
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Two positive numbers <math>x</math> and <math>y</math> are in the ratio <math>a: b</math> where <math>0 < a < b</math>. If <math>x+y = c</math>, then the smaller of <math>x</math> and <math>y</math> is | Two positive numbers <math>x</math> and <math>y</math> are in the ratio <math>a: b</math> where <math>0 < a < b</math>. If <math>x+y = c</math>, then the smaller of <math>x</math> and <math>y</math> is | ||
− | <math>\ | + | <math>\textbf{(A)} \ \frac{ac}{b} \qquad |
− | \ | + | \textbf{(B)} \ \frac{bc-ac}{b} \qquad |
− | \ | + | \textbf{(C)} \ \frac{ac}{a+b} \qquad |
− | \ | + | \textbf{(D)}\ \frac{bc}{a+b}\qquad |
− | \ | + | \textbf{(E)}\ \frac{ac}{b-a} </math> |
[[1982 AHSME Problems/Problem 5|Solution]] | [[1982 AHSME Problems/Problem 5|Solution]] | ||
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The sum of all but one of the interior angles of a convex polygon equals <math>2570^\circ</math>. The remaining angle is | The sum of all but one of the interior angles of a convex polygon equals <math>2570^\circ</math>. The remaining angle is | ||
− | <math>\ | + | <math>\textbf{(A)} \ 90^\circ \qquad |
− | \ | + | \textbf{(B)} \ 105^\circ \qquad |
− | \ | + | \textbf{(C)} \ 120^\circ \qquad |
− | \ | + | \textbf{(D)}\ 130^\circ\qquad |
− | \ | + | \textbf{(E)}\ 144^\circ </math> |
[[1982 AHSME Problems/Problem 6|Solution]] | [[1982 AHSME Problems/Problem 6|Solution]] | ||
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If the operation <math>x \star y</math> is defined by <math>x \star y = (x+1)(y+1) - 1</math>, then which one of the following is FALSE? | If the operation <math>x \star y</math> is defined by <math>x \star y = (x+1)(y+1) - 1</math>, then which one of the following is FALSE? | ||
− | <math>\ | + | <math>\textbf{(A)} \ x \star y = y\star x \text{ for all real } x,y. \\ |
− | \ | + | \textbf{(B)} \ x \star (y + z) = ( x \star y ) + (x \star z) \text{ for all real } x,y, \text{ and } z.\\ |
− | \ | + | \textbf{(C)} \ (x-1) \star (x+1) = (x \star x) - 1 \text{ for all real } x. \\ |
− | \ | + | \textbf{(D)} \ x \star 0 = x \text{ for all real } x. \\ |
− | \ | + | \textbf{(E)} \ x \star (y \star z) = (x \star y) \star z \text{ for all real } x,y, \text{ and } z. </math> |
[[1982 AHSME Problems/Problem 7|Solution]] | [[1982 AHSME Problems/Problem 7|Solution]] | ||
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[[1982 AHSME Problems/Problem 8|Solution]] | [[1982 AHSME Problems/Problem 8|Solution]] | ||
− | + | ||
== Problem 9 == | == Problem 9 == | ||
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The equation of the line is <math>x=</math> | The equation of the line is <math>x=</math> | ||
− | <math>\ | + | <math>\textbf {(A)}\ 2.5 \qquad |
− | \ | + | \textbf {(B)}\ 3.0 \qquad |
− | \ | + | \textbf {(C)}\ 3.5 \qquad |
− | \ | + | \textbf {(D)}\ 4.0\qquad |
− | \ | + | \textbf {(E)}\ 4.5 </math> |
[[1982 AHSME Problems/Problem 9|Solution]] | [[1982 AHSME Problems/Problem 9|Solution]] | ||
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label("$O$", O, dir(90));</asy> | label("$O$", O, dir(90));</asy> | ||
− | <math>\ | + | <math>\textbf {(A)}\ 30 \qquad |
− | \ | + | \textbf {(B)}\ 33 \qquad |
− | \ | + | \textbf {(C)}\ 36 \qquad |
− | \ | + | \textbf {(D)}\ 39 \qquad |
− | \ | + | \textbf {(E)}\ 42 </math> |
[[1982 AHSME Problems/Problem 10|Solution]] | [[1982 AHSME Problems/Problem 10|Solution]] | ||
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the difference between the first digit and the last digit is <math>2</math>? | the difference between the first digit and the last digit is <math>2</math>? | ||
− | <math>\ | + | <math>\textbf {(A)}\ 672 \qquad |
− | \ | + | \textbf {(B)}\ 784 \qquad |
− | \ | + | \textbf {(C)}\ 840 \qquad |
− | \ | + | \textbf {(D)}\ 896 \qquad |
− | \ | + | \textbf {(E)}\ 1008</math> |
[[1982 AHSME Problems/Problem 11|Solution]] | [[1982 AHSME Problems/Problem 11|Solution]] | ||
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Let <math>f(x) = ax^7+bx^3+cx-5</math>, where <math>a,b</math> and <math>c</math> are constants. If <math>f(-7) = 7</math>, the <math>f(7)</math> equals | Let <math>f(x) = ax^7+bx^3+cx-5</math>, where <math>a,b</math> and <math>c</math> are constants. If <math>f(-7) = 7</math>, the <math>f(7)</math> equals | ||
− | <math>\ | + | <math>\textbf {(A)}\ -17 \qquad |
− | \ | + | \textbf {(B)}\ -7 \qquad |
− | \ | + | \textbf {(C)}\ 14 \qquad |
− | \ | + | \textbf {(D)}\ 21\qquad |
− | \ | + | \textbf {(E)}\ \text{not uniquely determined}</math> |
[[1982 AHSME Problems/Problem 12|Solution]] | [[1982 AHSME Problems/Problem 12|Solution]] | ||
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If <math>a>1, b>1</math>, and <math>p=\frac{\log_b(\log_ba)}{\log_ba}</math>, then <math>a^p</math> equals | If <math>a>1, b>1</math>, and <math>p=\frac{\log_b(\log_ba)}{\log_ba}</math>, then <math>a^p</math> equals | ||
− | <math>\ | + | <math>\textbf {(A)}\ 1 \qquad |
− | \ | + | \textbf {(B)}\ b \qquad |
− | \ | + | \textbf {(C)}\ \log_ab \qquad |
− | \ | + | \textbf {(D)}\ \log_ba \qquad |
− | \ | + | \textbf {(E)}\ a^{\log_ba} </math> |
[[1982 AHSME Problems/Problem 13|Solution]] | [[1982 AHSME Problems/Problem 13|Solution]] | ||
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label("$G$", G, dir(100));</asy> | label("$G$", G, dir(100));</asy> | ||
− | <math>\ | + | <math>\textbf {(A)}\ 20 \qquad |
− | \ | + | \textbf {(B)}\ 15\sqrt{2} \qquad |
− | \ | + | \textbf {(C)}\ 24 \qquad |
− | \ | + | \textbf {(D)}\ 25 \qquad |
− | \ | + | \textbf {(E)}\ \text{none of these}</math> |
[[1982 AHSME Problems/Problem 14|Solution]] | [[1982 AHSME Problems/Problem 14|Solution]] | ||
− | + | ||
== Problem 15 == | == Problem 15 == | ||
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If <math>x</math> is not an integer, then <math>x+y</math> is | If <math>x</math> is not an integer, then <math>x+y</math> is | ||
− | <math>\ | + | <math>\textbf{(A) } \ \text{ an integer} \qquad |
− | \ | + | \textbf{(B) } \ \text{ between 4 and 5} \qquad |
− | \ | + | \textbf{(C) } \ \text{ between -4 and 4}\qquad\\ |
− | \ | + | \textbf{(D) } \ \text{ between 15 and 16}\qquad |
− | \ | + | \textbf{(E) } \ 16.5 </math> |
[[1982 AHSME Problems/Problem 15|Solution]] | [[1982 AHSME Problems/Problem 15|Solution]] | ||
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The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is | The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is | ||
− | <math>\ | + | <math>\textbf {(A)} \ 54 \qquad |
− | \ | + | \textbf {(B)} \ 72 \qquad |
− | \ | + | \textbf {(C)} \ 76 \qquad |
− | \ | + | \textbf {(D)} \ 84\qquad |
− | \ | + | \textbf {(E)} \ 86 </math> |
[[1982 AHSME Problems/Problem 16|Solution]] | [[1982 AHSME Problems/Problem 16|Solution]] | ||
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How many real numbers <math>x</math> satisfy the equation <math>3^{2x+2}-3^{x+3}-3^{x}+3=0</math>? | How many real numbers <math>x</math> satisfy the equation <math>3^{2x+2}-3^{x+3}-3^{x}+3=0</math>? | ||
− | <math>\ | + | <math>\textbf {(A)}\ 0 \qquad |
− | \ | + | \textbf {(B)}\ 1 \qquad |
− | \ | + | \textbf {(C)}\ 2 \qquad |
− | \ | + | \textbf {(D)}\ 3 \qquad |
− | \ | + | \textbf {(E)}\ 4 </math> |
[[1982 AHSME Problems/Problem 17|Solution]] | [[1982 AHSME Problems/Problem 17|Solution]] | ||
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label("$60^\circ$", H60, dir(25), fontsize(8));</asy> | label("$60^\circ$", H60, dir(25), fontsize(8));</asy> | ||
− | <math>\ | + | <math>\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad |
− | \ | + | \textbf{(B)}\ \frac{\sqrt{2}}{6} \qquad |
− | \ | + | \textbf{(C)}\ \frac{\sqrt{6}}{3} \qquad |
− | \ | + | \textbf{(D)}\ \frac{\sqrt{6}}{4}\qquad |
− | \ | + | \textbf{(E)}\ \frac{\sqrt{6}-\sqrt{2}}{4} </math> |
[[1982 AHSME Problems/Problem 18|Solution]] | [[1982 AHSME Problems/Problem 18|Solution]] | ||
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Let <math>f(x)=|x-2|+|x-4|-|2x-6|</math> for <math>2 \leq x\leq 8</math>. The sum of the largest and smallest values of <math>f(x)</math> is | Let <math>f(x)=|x-2|+|x-4|-|2x-6|</math> for <math>2 \leq x\leq 8</math>. The sum of the largest and smallest values of <math>f(x)</math> is | ||
− | <math>\ | + | <math>\textbf {(A)}\ 1 \qquad |
− | \ | + | \textbf {(B)}\ 2 \qquad |
− | \ | + | \textbf {(C)}\ 4 \qquad |
− | \ | + | \textbf {(D)}\ 6 \qquad |
− | \ | + | \textbf {(E)}\ \text{none of these} </math> |
[[1982 AHSME Problems/Problem 19|Solution]] | [[1982 AHSME Problems/Problem 19|Solution]] | ||
− | + | ||
== Problem 20 == | == Problem 20 == | ||
The number of pairs of positive integers <math>(x,y)</math> which satisfy the equation <math>x^2+y^2=x^3</math> is | The number of pairs of positive integers <math>(x,y)</math> which satisfy the equation <math>x^2+y^2=x^3</math> is | ||
− | <math>\ | + | <math>\textbf {(A)}\ 0 \qquad |
− | \ | + | \textbf {(B)}\ 1 \qquad |
− | \ | + | \textbf {(C)}\ 2 \qquad |
− | \ | + | \textbf {(D)}\ \text{not finite} \qquad |
− | \ | + | \textbf {(E)}\ \text{none of these} </math> |
[[1982 AHSME Problems/Problem 20|Solution]] | [[1982 AHSME Problems/Problem 20|Solution]] | ||
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draw(M--C--A--B--C^^B--N); | draw(M--C--A--B--C^^B--N); | ||
pair point=P; | pair point=P; | ||
− | markscalefactor=0. | + | markscalefactor=0.01; |
+ | draw(rightanglemark(B,C,N)); | ||
draw(rightanglemark(C,P,B)); | draw(rightanglemark(C,P,B)); | ||
label("$A$", A, dir(point--A)); | label("$A$", A, dir(point--A)); | ||
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label("$M$", M, S); | label("$M$", M, S); | ||
label("$N$", N, dir(C--A)*dir(90)); | label("$N$", N, dir(C--A)*dir(90)); | ||
− | label("$s$", B--C, NW);</asy> | + | label("$s$", B--C, NW); |
+ | </asy> | ||
− | <math>\ | + | <math>\textbf{(A)}\ s\sqrt 2 \qquad |
− | \ | + | \textbf{(B)}\ \frac 32s\sqrt2 \qquad |
− | \ | + | \textbf{(C)}\ 2s\sqrt2 \qquad |
− | \ | + | \textbf{(D)}\ \frac{s\sqrt5}{2}\qquad |
− | \ | + | \textbf{(E)}\ \frac{s\sqrt6}{2}</math> |
[[1982 AHSME Problems/Problem 21|Solution]] | [[1982 AHSME Problems/Problem 21|Solution]] | ||
− | + | ||
== Problem 22 == | == Problem 22 == | ||
− | In a narrow alley of width <math>w</math> a ladder of length a is placed with its foot at point P between the walls. | + | In a narrow alley of width <math>w</math> a ladder of length <math>a</math> is placed with its foot at point <math>P</math> between the walls. |
− | Resting against one wall at <math>Q</math>, the distance k above the ground makes a <math>45^\circ</math> angle with the ground. | + | Resting against one wall at <math>Q</math>, the distance <math>k</math> above the ground makes a <math>45^\circ</math> angle with the ground. |
− | Resting against the other wall at <math>R</math>, a distance h above the ground, the ladder makes a <math>75^\circ</math> angle with the ground. | + | Resting against the other wall at <math>R</math>, a distance <math>h</math> above the ground, the ladder makes a <math>75^\circ</math> angle with the ground. The width <math>w</math> is equal to |
− | The width <math>w</math> is equal to | ||
− | <math> \ | + | <math> \textbf{(A)}\ a\qquad |
− | \ | + | \textbf{(B)}\ RQ\qquad |
− | \ | + | \textbf{(C)}\ k\qquad |
− | \ | + | \textbf{(D)}\ \frac{h+k}{2}\qquad |
− | \ | + | \textbf{(E)}\ h </math> |
[[1982 AHSME Problems/Problem 22|Solution]] | [[1982 AHSME Problems/Problem 22|Solution]] | ||
− | + | ||
== Problem 23 == | == Problem 23 == | ||
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The cosine of the smallest angle is | The cosine of the smallest angle is | ||
− | <math> \ | + | <math> \textbf{(A)}\ \frac{3}{4}\qquad |
− | \ | + | \textbf{(B)}\ \frac{7}{10}\qquad |
− | \ | + | \textbf{(C)}\ \frac{2}{3}\qquad |
− | \ | + | \textbf{(D)}\ \frac{9}{14}\qquad |
− | \ | + | \textbf{(E)}\ \text{none of these} </math> |
[[1982 AHSME Problems/Problem 23|Solution]] | [[1982 AHSME Problems/Problem 23|Solution]] | ||
− | + | ||
== Problem 24 == | == Problem 24 == | ||
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label("7", H--J, dir(-30));</asy> | label("7", H--J, dir(-30));</asy> | ||
− | <math>\ | + | <math>\textbf {(A)}\ 2\sqrt{22} \qquad |
− | \ | + | \textbf {(B)}\ 7\sqrt{3} \qquad |
− | \ | + | \textbf {(C)}\ 9 \qquad |
− | \ | + | \textbf {(D)}\ 10 \qquad |
− | \ | + | \textbf {(E)}\ 13</math> |
[[1982 AHSME Problems/Problem 24|Solution]] | [[1982 AHSME Problems/Problem 24|Solution]] | ||
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== Problem 25 == | == Problem 25 == | ||
− | The adjacent map is part of a city: the small rectangles are | + | The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets. |
Each morning, a student walks from intersection <math>A</math> to intersection <math>B</math>, always walking along streets shown, | Each morning, a student walks from intersection <math>A</math> to intersection <math>B</math>, always walking along streets shown, | ||
and always going east or south. For variety, at each intersection where he has a choice, he chooses with | and always going east or south. For variety, at each intersection where he has a choice, he chooses with | ||
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label("S", (28,9.5), S);</asy> | label("S", (28,9.5), S);</asy> | ||
− | <math> \ | + | <math> \textbf{(A)}\ \frac{11}{32}\qquad |
− | \ | + | \textbf{(B)}\ \frac{1}{2}\qquad |
− | \ | + | \textbf{(C)}\ \frac{4}{7}\qquad |
− | \ | + | \textbf{(D)}\ \frac{21}{32}\qquad |
− | \ | + | \textbf{(E)}\ \frac{3}{4} </math> |
+ | |||
+ | [[1982 AHSME Problems/Problem 25|Solution]] | ||
− | |||
− | |||
== Problem 26 == | == Problem 26 == | ||
If the base <math>8</math> representation of a perfect square is <math>ab3c</math>, where <math>a\ne 0</math>, then <math>c</math> equals | If the base <math>8</math> representation of a perfect square is <math>ab3c</math>, where <math>a\ne 0</math>, then <math>c</math> equals | ||
− | <math>\ | + | <math>\textbf{(A)}\ 0\qquad |
− | \ | + | \textbf{(B)}\ 1 \qquad |
− | \ | + | \textbf{(C)}\ 3\qquad |
− | \ | + | \textbf{(D)}\ 4\qquad |
− | \ | + | \textbf{(E)}\ \text{not uniquely determined} </math> |
+ | |||
+ | [[1982 AHSME Problems/Problem 26|Solution]] | ||
− | |||
− | |||
== Problem 27 == | == Problem 27 == | ||
− | Suppose <math>z=a+bi</math> is a solution of the polynomial equation < | + | Suppose <math>z=a+bi</math> is a solution of the polynomial equation <cmath>c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0,</cmath> where <math>c_0, c_1, c_2, c_3, a,</math> and <math>b</math> are real constants and <math>i^2=-1.</math> Which of the following must also be a solution? |
− | are real constants and <math>i^2=-1</math> | ||
− | <math>\ | + | <math>\textbf{(A)}\ -a-bi\qquad |
− | \ | + | \textbf{(B)}\ a-bi\qquad |
− | \ | + | \textbf{(C)}\ -a+bi\qquad |
− | \ | + | \textbf{(D)}\ b+ai \qquad |
− | \ | + | \textbf{(E)}\ \text{none of these} </math> |
[[1982 AHSME Problems/Problem 27|Solution]] | [[1982 AHSME Problems/Problem 27|Solution]] | ||
− | + | ||
== Problem 28 == | == Problem 28 == | ||
− | A set of consecutive positive integers beginning with <math>1</math> is written on a blackboard. | + | A set of consecutive positive integers beginning with <math>1</math> is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is <math>35\frac{7}{17}</math>. What number was erased? |
− | One number is erased. The average (arithmetic mean) of the remaining numbers is <math>35\frac{7}{17}</math>. What number was erased? | ||
− | <math>\ | + | <math>\textbf{(A)}\ 6\qquad |
− | \ | + | \textbf{(B)}\ 7 \qquad |
− | \ | + | \textbf{(C)}\ 8 \qquad |
− | \ | + | \textbf{(D)}\ 9\qquad |
− | \ | + | \textbf{(E)}\ \text{cannot be determined}</math> |
[[1982 AHSME Problems/Problem 28|Solution]] | [[1982 AHSME Problems/Problem 28|Solution]] | ||
− | + | ||
== Problem 29 == | == Problem 29 == | ||
− | Let <math>x,y</math> | + | Let <math>x,y,</math> and <math>z</math> be three positive real numbers whose sum is <math>1.</math> If no one of these numbers is more than twice any other, |
then the minimum possible value of the product <math>xyz</math> is | then the minimum possible value of the product <math>xyz</math> is | ||
Line 476: | Line 476: | ||
[[1982 AHSME Problems/Problem 29|Solution]] | [[1982 AHSME Problems/Problem 29|Solution]] | ||
− | + | ||
== Problem 30 == | == Problem 30 == | ||
− | Find the units digit of the decimal expansion of | + | Find the units digit of the decimal expansion of <cmath>\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}.</cmath> |
− | |||
− | < | ||
<math>\textbf{(A)}\ 0\qquad | <math>\textbf{(A)}\ 0\qquad |
Latest revision as of 19:05, 11 September 2023
1982 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
When the polynomial is divided by the polynomial
, the remainder is
Problem 2
If a number eight times as large as is increased by two, then one fourth of the result equals
Problem 3
Evaluate at
.
Problem 4
The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is
Problem 5
Two positive numbers and
are in the ratio
where
. If
, then the smaller of
and
is
Problem 6
The sum of all but one of the interior angles of a convex polygon equals . The remaining angle is
Problem 7
If the operation is defined by
, then which one of the following is FALSE?
Problem 8
By definition, and
, where
are positive integers and
.
If
form an arithmetic progression with
, then
equals
Problem 9
A vertical line divides the triangle with vertices , and
in the
into two regions of equal area.
The equation of the line is
Problem 10
In the adjoining diagram, bisects
,
bisects
, and
is parallel to
.
If
, and
, then the perimeter of
is
Problem 11
How many integers with four different digits are there between and
such that the absolute value of
the difference between the first digit and the last digit is
?
Problem 12
Let , where
and
are constants. If
, the
equals
Problem 13
If , and
, then
equals
Problem 14
In the adjoining figure, points and
lie on line segment
, and
, and
are diameters of circle
, and
, respectively. Circles
, and
all have radius
and the line
is tangent to circle
at
.
If
intersects circle
at points
and
, then chord
has length
Problem 15
Let denote the greatest integer not exceeding
. Let
and
satisfy the simultaneous equations
If is not an integer, then
is
Problem 16
A wooden cube has edges of length meters. Square holes, of side one meter, centered in each face are cut through to the opposite face.
The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is
Problem 17
How many real numbers satisfy the equation
?
Problem 18
In the adjoining figure of a rectangular solid, and
. Find the cosine of
.
Problem 19
Let for
. The sum of the largest and smallest values of
is
Problem 20
The number of pairs of positive integers which satisfy the equation
is
Problem 21
In the adjoining figure, the triangle is a right triangle with
. Median
is perpendicular to median
,
and side
. The length of
is
Problem 22
In a narrow alley of width a ladder of length
is placed with its foot at point
between the walls.
Resting against one wall at
, the distance
above the ground makes a
angle with the ground.
Resting against the other wall at
, a distance
above the ground, the ladder makes a
angle with the ground. The width
is equal to
Problem 23
The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is
Problem 24
In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If ,
and
, then
equals
Problem 25
The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets.
Each morning, a student walks from intersection to intersection
, always walking along streets shown,
and always going east or south. For variety, at each intersection where he has a choice, he chooses with
probability
whether to go east or south. Find the probability that through any given morning, he goes through
.
Problem 26
If the base representation of a perfect square is
, where
, then
equals
Problem 27
Suppose is a solution of the polynomial equation
where
and
are real constants and
Which of the following must also be a solution?
Problem 28
A set of consecutive positive integers beginning with is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is
. What number was erased?
Problem 29
Let and
be three positive real numbers whose sum is
If no one of these numbers is more than twice any other,
then the minimum possible value of the product
is
Problem 30
Find the units digit of the decimal expansion of
See also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1981 AHSME |
Followed by 1983 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.