Difference between revisions of "1990 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1990 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
If <math>\dfrac{\frac{x}{4}}{2}=\dfrac{4}{\frac{x}{2}}</math>, then <math>x=</math> | If <math>\dfrac{\frac{x}{4}}{2}=\dfrac{4}{\frac{x}{2}}</math>, then <math>x=</math> | ||
− | <math>\ | + | <math>\textbf{(A)}\ \pm\frac{1}{2}\qquad\textbf{(B)}\ \pm 1\qquad\textbf{(C)}\ \pm 2\qquad\textbf{(D)}\ \pm 4\qquad\textbf{(E)}\ \pm 8</math> |
− | |||
[[1990 AHSME Problems/Problem 1|Solution]] | [[1990 AHSME Problems/Problem 1|Solution]] | ||
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<math>\left(\frac{1}{4}\right)^{-\tfrac{1}{4}}=</math> | <math>\left(\frac{1}{4}\right)^{-\tfrac{1}{4}}=</math> | ||
− | <math>\ | + | <math>\textbf{(A) } -16\qquad |
− | \ | + | \textbf{(B) } -\sqrt{2}\qquad |
− | \ | + | \textbf{(C) } -\frac{1}{16}\qquad |
− | \ | + | \textbf{(D) } \frac{1}{256}\qquad |
− | \ | + | \textbf{(E) } \sqrt{2}</math> |
[[1990 AHSME Problems/Problem 2|Solution]] | [[1990 AHSME Problems/Problem 2|Solution]] | ||
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The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is <math>75^\circ</math>, then the largest angle is | The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is <math>75^\circ</math>, then the largest angle is | ||
− | <math>\ | + | <math>\textbf{(A) } 95^\circ\qquad |
− | \ | + | \textbf{(B) } 100^\circ\qquad |
− | \ | + | \textbf{(C) } 105^\circ\qquad |
− | \ | + | \textbf{(D) } 110^\circ\qquad |
− | \ | + | \textbf{(E) } 115^\circ</math> |
− | |||
[[1990 AHSME Problems/Problem 3|Solution]] | [[1990 AHSME Problems/Problem 3|Solution]] | ||
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</asy> | </asy> | ||
− | Let <math>ABCD</math> be a parallelogram with <math>\angle{ABC}=120^\circ, AB= | + | Let <math>ABCD</math> be a parallelogram with <math>\angle{ABC}=120^\circ, AB=16</math> and <math>BC=10.</math> Extend <math>\overline{CD}</math> through <math>D</math> to <math>E</math> so that <math>DE=4.</math> If <math>\overline{BE}</math> intersects <math>\overline{AD}</math> at <math>F</math>, then <math>FD</math> is closest to |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
+ | <math>\textbf{(A) } 1\qquad | ||
+ | \textbf{(B) } 2\qquad | ||
+ | \textbf{(C) } 3\qquad | ||
+ | \textbf{(D) } 4\qquad | ||
+ | \textbf{(E) } 5</math> | ||
[[1990 AHSME Problems/Problem 4|Solution]] | [[1990 AHSME Problems/Problem 4|Solution]] | ||
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Which of these numbers is largest? | Which of these numbers is largest? | ||
− | <math>\ | + | <math>\textbf{(A) } \sqrt{\sqrt[3]{5\cdot 6}}\qquad |
− | \ | + | \textbf{(B) } \sqrt{6\sqrt[3]{5}}\qquad |
− | \ | + | \textbf{(C) } \sqrt{5\sqrt[3]{6}}\qquad |
− | \ | + | \textbf{(D) } \sqrt[3]{5\sqrt{6}}\qquad |
− | \ | + | \textbf{(E) } \sqrt[3]{6\sqrt{5}}</math> |
− | |||
[[1990 AHSME Problems/Problem 5|Solution]] | [[1990 AHSME Problems/Problem 5|Solution]] | ||
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Points <math>A</math> and <math>B</math> are <math>5</math> units apart. How many lines in a given plane containing <math>A</math> and <math>B</math> are <math>2</math> units from <math>A</math> and <math>3</math> units from <math>B</math>? | Points <math>A</math> and <math>B</math> are <math>5</math> units apart. How many lines in a given plane containing <math>A</math> and <math>B</math> are <math>2</math> units from <math>A</math> and <math>3</math> units from <math>B</math>? | ||
− | <math>\ | + | <math>\textbf{(A) } 0\qquad |
− | \ | + | \textbf{(B) } 1\qquad |
− | \ | + | \textbf{(C) } 2\qquad |
− | \ | + | \textbf{(D) } 3\qquad |
− | \ | + | \textbf{(E) }\text{more than }3</math> |
− | |||
[[1990 AHSME Problems/Problem 6|Solution]] | [[1990 AHSME Problems/Problem 6|Solution]] | ||
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A triangle with integral sides has perimeter <math>8</math>. The area of the triangle is | A triangle with integral sides has perimeter <math>8</math>. The area of the triangle is | ||
− | <math>\ | + | <math>\textbf{(A) } 2\sqrt{2}\qquad |
− | \ | + | \textbf{(B) } \frac{16}{9}\sqrt{3}\qquad |
− | \ | + | \textbf{(C) } 2\sqrt{3} \qquad |
− | \ | + | \textbf{(D) } 4\qquad |
− | \ | + | \textbf{(E) } 4\sqrt{2}</math> |
[[1990 AHSME Problems/Problem 7|Solution]] | [[1990 AHSME Problems/Problem 7|Solution]] | ||
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is | is | ||
− | <math>\ | + | <math>\textbf{(A) } 0\qquad |
− | \ | + | \textbf{(B) } 1\qquad |
− | \ | + | \textbf{(C) } 2\qquad |
− | \ | + | \textbf{(D) } 3\qquad |
− | \ | + | \textbf{(E) } \text{more than } 3</math> |
[[1990 AHSME Problems/Problem 8|Solution]] | [[1990 AHSME Problems/Problem 8|Solution]] | ||
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Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is | Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is | ||
− | <math>\ | + | <math>\textbf{(A) } 2\qquad |
− | \ | + | \textbf{(B) } 3\qquad |
− | \ | + | \textbf{(C) } 4\qquad |
− | \ | + | \textbf{(D) } 5\qquad |
− | \ | + | \textbf{(E) } 6</math> |
[[1990 AHSME Problems/Problem 9|Solution]] | [[1990 AHSME Problems/Problem 9|Solution]] | ||
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An <math>11\times 11\times 11</math> wooden cube is formed by gluing together <math>11^3</math> unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | An <math>11\times 11\times 11</math> wooden cube is formed by gluing together <math>11^3</math> unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | ||
− | <math>\ | + | <math>\textbf{(A) } 328\qquad |
− | \ | + | \textbf{(B) } 329\qquad |
− | \ | + | \textbf{(C) } 330\qquad |
− | \ | + | \textbf{(D) } 331\qquad |
− | \ | + | \textbf{(E) } 332</math> |
[[1990 AHSME Problems/Problem 10|Solution]] | [[1990 AHSME Problems/Problem 10|Solution]] | ||
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How many positive integers less than <math>50</math> have an odd number of positive integer divisors? | How many positive integers less than <math>50</math> have an odd number of positive integer divisors? | ||
− | <math>\ | + | <math>\textbf{(A) } 3\qquad |
− | \ | + | \textbf{(B) } 5\qquad |
− | \ | + | \textbf{(C) } 7\qquad |
− | \ | + | \textbf{(D) } 9\qquad |
− | \ | + | \textbf{(E) } 11</math> |
[[1990 AHSME Problems/Problem 11|Solution]] | [[1990 AHSME Problems/Problem 11|Solution]] | ||
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Let <math>f</math> be the function defined by <math>f(x)=ax^2-\sqrt{2}</math> for some positive <math>a</math>. If <math>f(f(\sqrt{2}))=-\sqrt{2}</math> then <math>a=</math> | Let <math>f</math> be the function defined by <math>f(x)=ax^2-\sqrt{2}</math> for some positive <math>a</math>. If <math>f(f(\sqrt{2}))=-\sqrt{2}</math> then <math>a=</math> | ||
− | <math>\ | + | <math>\textbf{(A) } \frac{2-\sqrt{2}}{2}\qquad |
− | \ | + | \textbf{(B) } \frac{1}{2}\qquad |
− | \ | + | \textbf{(C) } 2-\sqrt{2}\qquad |
− | \ | + | \textbf{(D) } \frac{\sqrt{2}}{2}\qquad |
− | \ | + | \textbf{(E) } \frac{2+\sqrt{2}}{2}</math> |
[[1990 AHSME Problems/Problem 12|Solution]] | [[1990 AHSME Problems/Problem 12|Solution]] | ||
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6.STOP. | 6.STOP. | ||
− | <math>\ | + | <math>\textbf{(A) } 19\qquad |
− | \ | + | \textbf{(B) } 21\qquad |
− | \ | + | \textbf{(C) } 23\qquad |
− | \ | + | \textbf{(D) } 199\qquad |
− | \ | + | \textbf{(E) } 201</math> |
[[1990 AHSME Problems/Problem 13|Solution]] | [[1990 AHSME Problems/Problem 13|Solution]] | ||
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An acute isosceles triangle, <math>ABC</math>, is inscribed in a circle. Through <math>B</math> and <math>C</math>, tangents to the circle are drawn, meeting at point <math>D</math>. If <math>\angle{ABC}=\angle{ACB}=2\angle{D}</math> and <math>x</math> is the radian measure of <math>\angle{A}</math>, then <math>x=</math> | An acute isosceles triangle, <math>ABC</math>, is inscribed in a circle. Through <math>B</math> and <math>C</math>, tangents to the circle are drawn, meeting at point <math>D</math>. If <math>\angle{ABC}=\angle{ACB}=2\angle{D}</math> and <math>x</math> is the radian measure of <math>\angle{A}</math>, then <math>x=</math> | ||
− | <math>\ | + | <math>\textbf{(A) } \frac{3\pi}{7}\qquad |
− | \ | + | \textbf{(B) } \frac{4\pi}{9}\qquad |
− | \ | + | \textbf{(C) } \frac{5\pi}{11}\qquad |
− | \ | + | \textbf{(D) } \frac{6\pi}{13}\qquad |
− | \ | + | \textbf{(E) } \frac{7\pi}{15}</math> |
[[1990 AHSME Problems/Problem 14|Solution]] | [[1990 AHSME Problems/Problem 14|Solution]] | ||
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Four whole numbers, when added three at a time, give the sums <math>180,197,208</math> and <math>222</math>. What is the largest of the four numbers? | Four whole numbers, when added three at a time, give the sums <math>180,197,208</math> and <math>222</math>. What is the largest of the four numbers? | ||
− | <math>\ | + | <math>\textbf{(A) } 77\qquad |
− | \ | + | \textbf{(B) } 83\qquad |
− | \ | + | \textbf{(C) } 89\qquad |
− | \ | + | \textbf{(D) } 95\qquad |
− | \ | + | \textbf{(E) cannot be determined from the given information}</math> |
[[1990 AHSME Problems/Problem 15|Solution]] | [[1990 AHSME Problems/Problem 15|Solution]] | ||
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At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If <math>13</math> married couples attended, how many handshakes were there among these <math>26</math> people? | At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If <math>13</math> married couples attended, how many handshakes were there among these <math>26</math> people? | ||
− | <math>\ | + | <math>\textbf{(A) } 78\qquad |
− | \ | + | \textbf{(B) } 185\qquad |
− | \ | + | \textbf{(C) } 234\qquad |
− | \ | + | \textbf{(D) } 312\qquad |
− | \ | + | \textbf{(E) } 325</math> |
− | |||
[[1990 AHSME Problems/Problem 16|Solution]] | [[1990 AHSME Problems/Problem 16|Solution]] | ||
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How many of the numbers, <math>100,101,\cdots,999</math> have three different digits in increasing order or in decreasing order? | How many of the numbers, <math>100,101,\cdots,999</math> have three different digits in increasing order or in decreasing order? | ||
− | <math>\ | + | <math>\textbf{(A) } 120\qquad |
− | \ | + | \textbf{(B) } 168\qquad |
− | \ | + | \textbf{(C) } 204\qquad |
− | \ | + | \textbf{(D) } 216\qquad |
− | \ | + | \textbf{(E) } 240</math> |
[[1990 AHSME Problems/Problem 17|Solution]] | [[1990 AHSME Problems/Problem 17|Solution]] | ||
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First <math>a</math> is chosen at random from the set <math>\{1,2,3,\cdots,99,100\}</math>, and then <math>b</math> is chosen at random from the same set. The probability that the integer <math>3^a+7^b</math> has units digit <math>8</math> is | First <math>a</math> is chosen at random from the set <math>\{1,2,3,\cdots,99,100\}</math>, and then <math>b</math> is chosen at random from the same set. The probability that the integer <math>3^a+7^b</math> has units digit <math>8</math> is | ||
− | <math>\ | + | <math>\textbf{(A) } \frac{1}{16}\qquad |
− | \ | + | \textbf{(B) } \frac{1}{8}\qquad |
− | \ | + | \textbf{(C) } \frac{3}{16}\qquad |
− | \ | + | \textbf{(D) } \frac{1}{5}\qquad |
− | \ | + | \textbf{(E) } \frac{1}{4}</math> |
[[1990 AHSME Problems/Problem 18|Solution]] | [[1990 AHSME Problems/Problem 18|Solution]] | ||
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For how many integers <math>N</math> between <math>1</math> and <math>1990</math> is the improper fraction <math>\frac{N^2+7}{N+4}</math> <math>\underline{not}</math> in lowest terms? | For how many integers <math>N</math> between <math>1</math> and <math>1990</math> is the improper fraction <math>\frac{N^2+7}{N+4}</math> <math>\underline{not}</math> in lowest terms? | ||
− | <math>\ | + | <math>\textbf{(A) } 0\qquad |
− | \ | + | \textbf{(B) } 86\qquad |
− | \ | + | \textbf{(C) } 90\qquad |
− | \ | + | \textbf{(D) } 104\qquad |
− | \ | + | \textbf{(E) } 105</math> |
[[1990 AHSME Problems/Problem 19|Solution]] | [[1990 AHSME Problems/Problem 19|Solution]] | ||
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</asy> | </asy> | ||
− | In the figure <math>ABCD</math> is a quadrilateral with right angles at <math>A</math> and <math>C</math>. Points <math>E</math> and <math>F</math> are on <math>\overline{AC}</math>, and <math>\overline{DE}</math> and <math>\overline{BF}</math> are | + | In the figure <math>ABCD</math> is a quadrilateral with right angles at <math>A</math> and <math>C</math>. Points <math>E</math> and <math>F</math> are on <math>\overline{AC}</math>, and <math>\overline{DE}</math> and <math>\overline{BF}</math> are perpendicular to <math>\overline{AC}</math>. If <math>AE=3, DE=5, </math> and <math>CE=7</math>, then <math>BF=</math> |
− | <math>\ | + | <math>\textbf{(A) } 3.6\qquad |
− | \ | + | \textbf{(B) } 4\qquad |
− | \ | + | \textbf{(C) } 4.2\qquad |
− | \ | + | \textbf{(D) } 4.5\qquad |
− | \ | + | \textbf{(E) } 5</math> |
[[1990 AHSME Problems/Problem 20|Solution]] | [[1990 AHSME Problems/Problem 20|Solution]] | ||
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Consider a pyramid <math>P-ABCD</math> whose base <math>ABCD</math> is square and whose vertex <math>P</math> is equidistant from <math>A,B,C</math> and <math>D</math>. If <math>AB=1</math> and <math>\angle{APB}=2\theta</math>, then the volume of the pyramid is | Consider a pyramid <math>P-ABCD</math> whose base <math>ABCD</math> is square and whose vertex <math>P</math> is equidistant from <math>A,B,C</math> and <math>D</math>. If <math>AB=1</math> and <math>\angle{APB}=2\theta</math>, then the volume of the pyramid is | ||
− | <math>\ | + | <math>\textbf{(A) } \frac{\sin(\theta)}{6}\qquad |
− | \ | + | \textbf{(B) } \frac{\cot(\theta)}{6}\qquad |
− | \ | + | \textbf{(C) } \frac{1}{6\sin(\theta)}\qquad |
− | \ | + | \textbf{(D) } \frac{1-\sin(2\theta)}{6}\qquad |
− | \ | + | \textbf{(E) } \frac{\sqrt{\cos(2\theta)}}{6\sin(\theta)}</math> |
− | |||
[[1990 AHSME Problems/Problem 21|Solution]] | [[1990 AHSME Problems/Problem 21|Solution]] | ||
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If the six solutions of <math>x^6=-64</math> are written in the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are real, then the product of those solutions with <math>a>0</math> is | If the six solutions of <math>x^6=-64</math> are written in the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are real, then the product of those solutions with <math>a>0</math> is | ||
− | + | <math>\textbf{(A) } -2\qquad | |
− | <math>\ | + | \textbf{(B) } 0\qquad |
− | \ | + | \textbf{(C) } 2i\qquad |
− | \ | + | \textbf{(D) } 4\qquad |
− | \ | + | \textbf{(E) } 16</math> |
− | \ | ||
[[1990 AHSME Problems/Problem 22|Solution]] | [[1990 AHSME Problems/Problem 22|Solution]] | ||
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== Problem 23 == | == Problem 23 == | ||
− | If <math>x,y>0, log_y(x)+log_x(y)=\frac{10}{3} \text{ and } xy=144,\text{ then }\frac{x+y}{2}=</math> | + | If <math>x,y>0, \log_y(x)+\log_x(y)=\frac{10}{3} \text{ and } xy=144,\text{ then }\frac{x+y}{2}=</math> |
− | |||
− | <math>\ | + | <math>\textbf{(A) } 12\sqrt{2}\qquad |
− | \ | + | \textbf{(B) } 13\sqrt{3}\qquad |
− | \ | + | \textbf{(C) } 24\qquad |
− | \ | + | \textbf{(D) } 30\qquad |
− | \ | + | \textbf{(E) } 36</math> |
[[1990 AHSME Problems/Problem 23|Solution]] | [[1990 AHSME Problems/Problem 23|Solution]] | ||
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All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined? | All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined? | ||
− | + | <cmath>\begin{tabular}[t]{|c|c|c|c|} | |
− | |||
− | < | ||
\multicolumn{4}{c}{Average Scores}\\\hline | \multicolumn{4}{c}{Average Scores}\\\hline | ||
Category&Adams&Baker&Adams\&Baker\\\hline | Category&Adams&Baker&Adams\&Baker\\\hline | ||
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Girls&76&90&?\\ | Girls&76&90&?\\ | ||
Boys\&Girls&74&84& \\\hline | Boys\&Girls&74&84& \\\hline | ||
− | \end{tabular}</ | + | \end{tabular}</cmath> |
− | + | <math>\textbf{(A) } 81\qquad | |
− | + | \textbf{(B) } 82\qquad | |
− | + | \textbf{(C) } 83\qquad | |
− | <math>\ | + | \textbf{(D) } 84\qquad |
− | \ | + | \textbf{(E) } 85</math> |
− | \ | ||
− | \ | ||
− | \ | ||
[[1990 AHSME Problems/Problem 24|Solution]] | [[1990 AHSME Problems/Problem 24|Solution]] | ||
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Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? | Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? | ||
− | <math>\ | + | <math>\textbf{(A) } 1-\frac{\sqrt{3}}{2}\qquad |
− | \ | + | \textbf{(B) } \frac{2\sqrt{3}-3}{2}\qquad |
− | \ | + | \textbf{(C) } \frac{\sqrt{2}}{6}\qquad |
− | \ | + | \textbf{(D) } \frac{1}{4}\qquad |
− | \ | + | \textbf{(E) } \frac{\sqrt{3}(2-\sqrt{2})}{4}</math> |
[[1990 AHSME Problems/Problem 25|Solution]] | [[1990 AHSME Problems/Problem 25|Solution]] | ||
== Problem 26 == | == Problem 26 == | ||
+ | Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (<i>not</i> the original number the person picked.) | ||
+ | <asy> | ||
+ | unitsize(2 cm); | ||
− | + | for(int i = 1; i <= 10; ++i) { | |
+ | label("``" + (string) i + "''", dir(90 - 360/10*(i - 1))); | ||
+ | } | ||
+ | </asy> | ||
+ | The number picked by the person who announced the average <math>6</math> was | ||
− | (A) 1 (B) 5 (C) 6 (D) 10 (E) not uniquely determined from the given information | + | <math>\textbf{(A) } 1 \qquad |
+ | \textbf{(B) } 5 \qquad | ||
+ | \textbf{(C) } 6 \qquad | ||
+ | \textbf{(D) } 10 \qquad | ||
+ | \textbf{(E) }\text{not uniquely determined from the given information}</math> | ||
[[1990 AHSME Problems/Problem 26|Solution]] | [[1990 AHSME Problems/Problem 26|Solution]] | ||
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Which of these triples could <math>\underline{not}</math> be the lengths of the three altitudes of a triangle? | Which of these triples could <math>\underline{not}</math> be the lengths of the three altitudes of a triangle? | ||
− | <math>\ | + | <math>\textbf{(A) } 1,\sqrt{3},2\qquad |
− | \ | + | \textbf{(B) } 3,4,5\qquad |
− | \ | + | \textbf{(C) } 5,12,13\qquad |
− | \ | + | \textbf{(D) } 7,8,\sqrt{113}\qquad |
− | \ | + | \textbf{(E) } 8,15,17</math> |
[[1990 AHSME Problems/Problem 27|Solution]] | [[1990 AHSME Problems/Problem 27|Solution]] | ||
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A quadrilateral that has consecutive sides of lengths <math>70,90,130</math> and <math>110</math> is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length <math>x</math> and <math>y</math>. Find <math>|x-y|</math>. | A quadrilateral that has consecutive sides of lengths <math>70,90,130</math> and <math>110</math> is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length <math>x</math> and <math>y</math>. Find <math>|x-y|</math>. | ||
− | <math>\ | + | <math>\textbf{(A) } 12\qquad |
− | \ | + | \textbf{(B) } 13\qquad |
− | \ | + | \textbf{(C) } 14\qquad |
− | \ | + | \textbf{(D) } 15\qquad |
− | \ | + | \textbf{(E) } 16</math> |
[[1990 AHSME Problems/Problem 28|Solution]] | [[1990 AHSME Problems/Problem 28|Solution]] | ||
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A subset of the integers <math>1,2,\cdots,100</math> has the property that none of its members is 3 times another. What is the largest number of members such a subset can have? | A subset of the integers <math>1,2,\cdots,100</math> has the property that none of its members is 3 times another. What is the largest number of members such a subset can have? | ||
− | <math>\ | + | <math>\textbf{(A) } 50\qquad |
− | \ | + | \textbf{(B) } 66\qquad |
− | \ | + | \textbf{(C) } 67\qquad |
− | \ | + | \textbf{(D) } 76\qquad |
− | \ | + | \textbf{(E) } 78</math> |
[[1990 AHSME Problems/Problem 29|Solution]] | [[1990 AHSME Problems/Problem 29|Solution]] | ||
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If <math>R_n=\frac{1}{2}(a^n+b^n)</math> where <math>a=3+2\sqrt{2}</math> and <math>b=3-2\sqrt{2}</math>, and <math>n=0,1,2,\cdots,</math> then <math>R_{12345}</math> is an integer. Its units digit is | If <math>R_n=\frac{1}{2}(a^n+b^n)</math> where <math>a=3+2\sqrt{2}</math> and <math>b=3-2\sqrt{2}</math>, and <math>n=0,1,2,\cdots,</math> then <math>R_{12345}</math> is an integer. Its units digit is | ||
− | <math>\ | + | <math>\textbf{(A) } 1\qquad |
− | \ | + | \textbf{(B) } 3\qquad |
− | \ | + | \textbf{(C) } 5\qquad |
− | \ | + | \textbf{(D) } 7\qquad |
− | \ | + | \textbf{(E) } 9</math> |
[[1990 AHSME Problems/Problem 30|Solution]] | [[1990 AHSME Problems/Problem 30|Solution]] |
Latest revision as of 00:15, 10 September 2021
1990 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
If , then
Problem 2
Problem 3
The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is , then the largest angle is
Problem 4
Let be a parallelogram with and Extend through to so that If intersects at , then is closest to
Problem 5
Which of these numbers is largest?
Problem 6
Points and are units apart. How many lines in a given plane containing and are units from and units from ?
Problem 7
A triangle with integral sides has perimeter . The area of the triangle is
Problem 8
The number of real solutions of the equation is
Problem 9
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is
Problem 10
An wooden cube is formed by gluing together unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
Problem 11
How many positive integers less than have an odd number of positive integer divisors?
Problem 12
Let be the function defined by for some positive . If then
Problem 13
If the following instructions are carried out by a computer, which value of will be printed because of instruction ?
1. START AT AND AT . 2. INCREASE THE VALUE OF BY . 3. INCREASE THE VALUE OF BY THE VALUE OF . 4. IF IS AT LEAST , THEN GO TO INSTRUCTION ; OTHERWISE, GO TO INSTRUCTION . AND PROCEED FROM THERE. 5. PRINT THE VALUE OF . 6.STOP.
Problem 14
An acute isosceles triangle, , is inscribed in a circle. Through and , tangents to the circle are drawn, meeting at point . If and is the radian measure of , then
Problem 15
Four whole numbers, when added three at a time, give the sums and . What is the largest of the four numbers?
Problem 16
At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If married couples attended, how many handshakes were there among these people?
Problem 17
How many of the numbers, have three different digits in increasing order or in decreasing order?
Problem 18
First is chosen at random from the set , and then is chosen at random from the same set. The probability that the integer has units digit is
Problem 19
For how many integers between and is the improper fraction in lowest terms?
Problem 20
In the figure is a quadrilateral with right angles at and . Points and are on , and and are perpendicular to . If and , then
Problem 21
Consider a pyramid whose base is square and whose vertex is equidistant from and . If and , then the volume of the pyramid is
Problem 22
If the six solutions of are written in the form , where and are real, then the product of those solutions with is
Problem 23
If
Problem 24
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
Problem 25
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?
Problem 26
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.) The number picked by the person who announced the average was
Problem 27
Which of these triples could be the lengths of the three altitudes of a triangle?
Problem 28
A quadrilateral that has consecutive sides of lengths and is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length and . Find .
Problem 29
A subset of the integers has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
Problem 30
If where and , and then is an integer. Its units digit is
See also
1990 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1989 AHSME |
Followed by 1991 AHSME | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.