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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>\text{(A)}\ \pm\frac{1}{2}\qquad\text{(B)}\ \pm 1\qquad\text{(C)}\ \pm 2\qquad\text{(D)}\ \pm 4\qquad\text{(E)}\ \pm 8</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>\text{(A) } -16\quad
+
<math>\textbf{(A) } -16\qquad
−
\text{(B) } -\sqrt{2}\quad
+
\textbf{(B) } -\sqrt{2}\qquad
−
\text{(C) } -\frac{1}{16}\quad
+
\textbf{(C) } -\frac{1}{16}\qquad
−
\text{(D) } \frac{1}{256}\quad
+
\textbf{(D) } \frac{1}{256}\qquad
−
\text{(E) } \sqrt{2}</math>
+
\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>\text{(A) } 95^\circ\quad
+
<math>\textbf{(A) } 95^\circ\qquad
−
\text{(B) } 100^\circ\quad
+
\textbf{(B) } 100^\circ\qquad
−
\text{(C) } 105^\circ\quad
+
\textbf{(C) } 105^\circ\qquad
−
\text{(D) } 110^\circ\quad
+
\textbf{(D) } 110^\circ\qquad
−
\text{(E) } 115^\circ</math>
+
\textbf{(E) } 115^\circ</math>
−
 
 
  
 
[[1990 AHSME Problems/Problem 3|Solution]]
 
[[1990 AHSME Problems/Problem 3|Solution]]
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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
 
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>\text{(A) } 1\quad
+
<math>\textbf{(A) } 1\qquad
−
\text{(B) } 2\quad
+
\textbf{(B) } 2\qquad
−
\text{(C) } 3\quad
+
\textbf{(C) } 3\qquad
−
\text{(D) } 4\quad
+
\textbf{(D) } 4\qquad
−
\text{(E) } 5</math>
+
\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>\text{(A) } \sqrt{\sqrt[3]{5\cdot 6}}\quad
+
<math>\textbf{(A) } \sqrt{\sqrt[3]{5\cdot 6}}\qquad
−
\text{(B) } \sqrt{6\sqrt[3]{5}}\quad
+
\textbf{(B) } \sqrt{6\sqrt[3]{5}}\qquad
−
\text{(C) } \sqrt{5\sqrt[3]{6}}\quad
+
\textbf{(C) } \sqrt{5\sqrt[3]{6}}\qquad
−
\text{(D) } \sqrt[3]{5\sqrt{6}}\quad
+
\textbf{(D) } \sqrt[3]{5\sqrt{6}}\qquad
−
\text{(E) } \sqrt[3]{6\sqrt{5}}</math>
+
\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>\text{(A) } 0\quad
+
<math>\textbf{(A) } 0\qquad
−
\text{(B) } 1\quad
+
\textbf{(B) } 1\qquad
−
\text{(C) } 2\quad
+
\textbf{(C) } 2\qquad
−
\text{(D) } 3\quad
+
\textbf{(D) } 3\qquad
−
\text{(E) more than }3</math>
+
\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>\text{(A) } 2\sqrt{2}\quad
+
<math>\textbf{(A) } 2\sqrt{2}\qquad
−
\text{(B) } \frac{16}{9}\sqrt{3}\quad
+
\textbf{(B) } \frac{16}{9}\sqrt{3}\qquad
−
\text{(C) }2\sqrt{3} \quad
+
\textbf{(C) } 2\sqrt{3} \qquad
−
\text{(D) } 4\quad
+
\textbf{(D) } 4\qquad
−
\text{(E) } 4\sqrt{2}</math>
+
\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>\text{(A) } 0\quad
+
<math>\textbf{(A) } 0\qquad
−
\text{(B) } 1\quad
+
\textbf{(B) } 1\qquad
−
\text{(C) } 2\quad
+
\textbf{(C) } 2\qquad
−
\text{(D) } 3\quad
+
\textbf{(D) } 3\qquad
−
\text{(E) more than } 3</math>
+
\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>\text{(A) } 2\quad
+
<math>\textbf{(A) } 2\qquad
−
\text{(B) } 3\quad
+
\textbf{(B) } 3\qquad
−
\text{(C) } 4\quad
+
\textbf{(C) } 4\qquad
−
\text{(D) } 5\quad
+
\textbf{(D) } 5\qquad
−
\text{(E) } 6</math>
+
\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>\text{(A) }  328\quad
+
<math>\textbf{(A) }  328\qquad
−
\text{(B) }  329\quad
+
\textbf{(B) }  329\qquad
−
\text{(C) }  330\quad
+
\textbf{(C) }  330\qquad
−
\text{(D) }  331\quad
+
\textbf{(D) }  331\qquad
−
\text{(E) }  332</math>
+
\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>\text{(A) } 3\quad
+
<math>\textbf{(A) } 3\qquad
−
\text{(B) } 5\quad
+
\textbf{(B) } 5\qquad
−
\text{(C) } 7\quad
+
\textbf{(C) } 7\qquad
−
\text{(D) } 9\quad
+
\textbf{(D) } 9\qquad
−
\text{(E) } 11</math>
+
\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>\text{(A) } \frac{2-\sqrt{2}}{2}\quad
+
<math>\textbf{(A) } \frac{2-\sqrt{2}}{2}\qquad
−
\text{(B) } \frac{1}{2}\quad
+
\textbf{(B) } \frac{1}{2}\qquad
−
\text{(C) } 2-\sqrt{2}\quad
+
\textbf{(C) } 2-\sqrt{2}\qquad
−
\text{(D) } \frac{\sqrt{2}}{2}\quad
+
\textbf{(D) } \frac{\sqrt{2}}{2}\qquad
−
\text{(E) } \frac{2+\sqrt{2}}{2}</math>
+
\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>\text{(A) } 19\quad
+
<math>\textbf{(A) } 19\qquad
−
\text{(B) } 21\quad
+
\textbf{(B) } 21\qquad
−
\text{(C) } 23\quad
+
\textbf{(C) } 23\qquad
−
\text{(D) } 199\quad
+
\textbf{(D) } 199\qquad
−
\text{(E) } 201</math>
+
\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>\text{(A) } \frac{3\pi}{7}\quad
+
<math>\textbf{(A) } \frac{3\pi}{7}\qquad
−
\text{(B) } \frac{4\pi}{9}\quad
+
\textbf{(B) } \frac{4\pi}{9}\qquad
−
\text{(C) } \frac{5\pi}{11}\quad
+
\textbf{(C) } \frac{5\pi}{11}\qquad
−
\text{(D) } \frac{6\pi}{13}\quad
+
\textbf{(D) } \frac{6\pi}{13}\qquad
−
\text{(E) } \frac{7\pi}{15}</math>
+
\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>\text{(A) } 77\quad
+
<math>\textbf{(A) } 77\qquad
−
\text{(B) } 83\quad
+
\textbf{(B) } 83\qquad
−
\text{(C) } 89\quad
+
\textbf{(C) } 89\qquad
−
\text{(D) } 95\quad
+
\textbf{(D) } 95\qquad
−
\text{(E) cannot be determined from the given information}</math>
+
\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>\text{(A) } 78\quad
+
<math>\textbf{(A) } 78\qquad
−
\text{(B) } 185\quad
+
\textbf{(B) } 185\qquad
−
\text{(C) } 234\quad
+
\textbf{(C) } 234\qquad
−
\text{(D) } 312\quad
+
\textbf{(D) } 312\qquad
−
\text{(E) } 325</math>
+
\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>\text{(A) } 120\quad
+
<math>\textbf{(A) } 120\qquad
−
\text{(B) } 168\quad
+
\textbf{(B) } 168\qquad
−
\text{(C) } 204\quad
+
\textbf{(C) } 204\qquad
−
\text{(D) } 216\quad
+
\textbf{(D) } 216\qquad
−
\text{(E) } 240</math>
+
\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>\text{(A) } \frac{1}{16}\quad
+
<math>\textbf{(A) } \frac{1}{16}\qquad
−
\text{(B) } \frac{1}{8}\quad
+
\textbf{(B) } \frac{1}{8}\qquad
−
\text{(C) } \frac{3}{16}\quad
+
\textbf{(C) } \frac{3}{16}\qquad
−
\text{(D) } \frac{1}{5}\quad
+
\textbf{(D) } \frac{1}{5}\qquad
−
\text{(E) } \frac{1}{4}</math>
+
\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>\text{(A) } 0\quad
+
<math>\textbf{(A) } 0\qquad
−
\text{(B) } 86\quad
+
\textbf{(B) } 86\qquad
−
\text{(C) } 90\quad
+
\textbf{(C) } 90\qquad
−
\text{(D) } 104\quad
+
\textbf{(D) } 104\qquad
−
\text{(E) } 105</math>
+
\textbf{(E) } 105</math>
  
 
[[1990 AHSME Problems/Problem 19|Solution]]
 
[[1990 AHSME Problems/Problem 19|Solution]]
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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>
 
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>\text{(A) } 3.6\quad
+
<math>\textbf{(A) } 3.6\qquad
−
\text{(B) } 4\quad
+
\textbf{(B) } 4\qquad
−
\text{(C) } 4.2\quad
+
\textbf{(C) } 4.2\qquad
−
\text{(D) } 4.5\quad
+
\textbf{(D) } 4.5\qquad
−
\text{(E) } 5</math>
+
\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>\text{(A) } \frac{\sin(\theta)}{6}\quad
+
<math>\textbf{(A) } \frac{\sin(\theta)}{6}\qquad
−
\text{(B) } \frac{\cot(\theta)}{6}\quad
+
\textbf{(B) } \frac{\cot(\theta)}{6}\qquad
−
\text{(C) } \frac{1}{6\sin(\theta)}\quad
+
\textbf{(C) } \frac{1}{6\sin(\theta)}\qquad
−
\text{(D) } \frac{1-\sin(2\theta)}{6}\quad
+
\textbf{(D) } \frac{1-\sin(2\theta)}{6}\qquad
−
\text{(E) } \frac{\sqrt{\cos(2\theta)}}{6\sin(\theta)}</math>
+
\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>\text{(A) } -2\quad
+
\textbf{(B) } 0\qquad
−
\text{(B) } 0\quad
+
\textbf{(C) } 2i\qquad
−
\text{(C) } 2i\quad
+
\textbf{(D) } 4\qquad
−
\text{(D) } 4\quad
+
\textbf{(E) } 16</math>
−
\text{(E) } 16</math>
 
  
 
[[1990 AHSME Problems/Problem 22|Solution]]
 
[[1990 AHSME Problems/Problem 22|Solution]]
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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>\textbf{(A) } 12\sqrt{2}\qquad
−
<math>\text{(A) } 12\sqrt{2}\quad
+
\textbf{(B) } 13\sqrt{3}\qquad
−
\text{(B) } 13\sqrt{3}\quad
+
\textbf{(C) } 24\qquad
−
\text{(C) } 24\quad
+
\textbf{(D) } 30\qquad
−
\text{(D) } 30\quad
+
\textbf{(E) } 36</math>
−
\text{(E) } 36</math>
 
  
 
[[1990 AHSME Problems/Problem 23|Solution]]
 
[[1990 AHSME Problems/Problem 23|Solution]]
Line 325: Line 319:
  
 
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|}
−
 
 
−
<math>\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
Line 333: Line 325:
 
Girls&76&90&?\\
 
Girls&76&90&?\\
 
Boys\&Girls&74&84& \\\hline
 
Boys\&Girls&74&84& \\\hline
−
\end{tabular}</math>
+
\end{tabular}</cmath>
−
 
+
<math>\textbf{(A) } 81\qquad
−
 
+
\textbf{(B) } 82\qquad
−
 
+
\textbf{(C) } 83\qquad
−
<math>\text{(A) } 81\quad
+
\textbf{(D) } 84\qquad
−
\text{(B) } 82\quad
+
\textbf{(E) } 85</math>
−
\text{(C) } 83\quad
 
−
\text{(D) } 84\quad
 
−
\text{(E) } 85</math>
 
  
 
[[1990 AHSME Problems/Problem 24|Solution]]
 
[[1990 AHSME Problems/Problem 24|Solution]]
Line 350: Line 339:
 
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>\text{(A) } 1-\frac{\sqrt{3}}{2}\quad
+
<math>\textbf{(A) } 1-\frac{\sqrt{3}}{2}\qquad
−
\text{(B) } \frac{2\sqrt{3}-3}{2}\quad
+
\textbf{(B) } \frac{2\sqrt{3}-3}{2}\qquad
−
\text{(C) } \frac{\sqrt{2}}{6}\quad
+
\textbf{(C) } \frac{\sqrt{2}}{6}\qquad
−
\text{(D) } \frac{1}{4}\quad
+
\textbf{(D) } \frac{1}{4}\qquad
−
\text{(E) } \frac{\sqrt{3}(2-\sqrt{2})}{4}</math>
+
\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);
  
−
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The average announced by each person was (in order around the circle) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (NOT the original number the person picked). The number picked by the person who announced the average 6 was
+
for(int i = 1; i <= 10; ++i) {
 +
  label("``" + (string) i + "&#039;&#039;", dir(90 - 360/10*(i - 1)));
 +
}
 +
</asy>
 +
The number picked by the person who announced the average <math>6</math> was
  
−
<math>\text{(A) } 1\quad
+
<math>\textbf{(A) } 1 \qquad
−
\text{(B) } 5\quad
+
\textbf{(B) } 5 \qquad
−
\text{(C) } 6\quad
+
\textbf{(C) } 6 \qquad
−
\text{(D) } 10\quad
+
\textbf{(D) } 10 \qquad
−
\text{(E) not uniquely determined from the given information} </math>
+
\textbf{(E) }\text{not uniquely determined from the given information}</math>
  
 
[[1990 AHSME Problems/Problem 26|Solution]]
 
[[1990 AHSME Problems/Problem 26|Solution]]
Line 374: Line 370:
 
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>\text{(A) } 1,\sqrt{3},2\quad
+
<math>\textbf{(A) } 1,\sqrt{3},2\qquad
−
\text{(B) } 3,4,5\quad
+
\textbf{(B) } 3,4,5\qquad
−
\text{(C) } 5,12,13\quad
+
\textbf{(C) } 5,12,13\qquad
−
\text{(D) } 7,8,\sqrt{113}\quad
+
\textbf{(D) } 7,8,\sqrt{113}\qquad
−
\text{(E) } 8,15,17</math>
+
\textbf{(E) } 8,15,17</math>
  
 
[[1990 AHSME Problems/Problem 27|Solution]]
 
[[1990 AHSME Problems/Problem 27|Solution]]
Line 386: Line 382:
 
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>\text{(A) } 12\quad
+
<math>\textbf{(A) } 12\qquad
−
\text{(B) } 13\quad
+
\textbf{(B) } 13\qquad
−
\text{(C) } 14\quad
+
\textbf{(C) } 14\qquad
−
\text{(D) } 15\quad
+
\textbf{(D) } 15\qquad
−
\text{(E) } 16</math>
+
\textbf{(E) } 16</math>
  
 
[[1990 AHSME Problems/Problem 28|Solution]]
 
[[1990 AHSME Problems/Problem 28|Solution]]
Line 398: Line 394:
 
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>\text{(A) } 50\quad
+
<math>\textbf{(A) } 50\qquad
−
\text{(B) } 66\quad
+
\textbf{(B) } 66\qquad
−
\text{(C) } 67\quad
+
\textbf{(C) } 67\qquad
−
\text{(D) } 76\quad
+
\textbf{(D) } 76\qquad
−
\text{(E) } 78</math>
+
\textbf{(E) } 78</math>
  
 
[[1990 AHSME Problems/Problem 29|Solution]]
 
[[1990 AHSME Problems/Problem 29|Solution]]
Line 410: Line 406:
 
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>\text{(A) } 1\quad
+
<math>\textbf{(A) } 1\qquad
−
\text{(B) } 3\quad
+
\textbf{(B) } 3\qquad
−
\text{(C) } 5\quad
+
\textbf{(C) } 5\qquad
−
\text{(D) } 7\quad
+
\textbf{(D) } 7\qquad
−
\text{(E) } 9</math>
+
\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: 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

If $\dfrac{\frac{x}{4}}{2}=\dfrac{4}{\frac{x}{2}}$, then $x=$

$\textbf{(A)}\ \pm\frac{1}{2}\qquad\textbf{(B)}\ \pm 1\qquad\textbf{(C)}\ \pm 2\qquad\textbf{(D)}\ \pm 4\qquad\textbf{(E)}\ \pm 8$

Solution

Problem 2

$\left(\frac{1}{4}\right)^{-\tfrac{1}{4}}=$

$\textbf{(A) } -16\qquad \textbf{(B) } -\sqrt{2}\qquad \textbf{(C) } -\frac{1}{16}\qquad \textbf{(D) } \frac{1}{256}\qquad \textbf{(E) } \sqrt{2}$

Solution

Problem 3

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ$, then the largest angle is

$\textbf{(A) } 95^\circ\qquad \textbf{(B) } 100^\circ\qquad \textbf{(C) } 105^\circ\qquad \textbf{(D) } 110^\circ\qquad \textbf{(E) } 115^\circ$

Solution

Problem 4

[asy] draw((0,0)--(16,0)--(21,5*sqrt(3))--(5,5*sqrt(3))--cycle,dot); draw((5,5*sqrt(3))--(1,5*sqrt(3))--(16,0),dot); MP("A",(0,0),S);MP("B",(16,0),S);MP("C",(21,5sqrt(3)),NE);MP("D",(5,5sqrt(3)),N);MP("E",(1,5sqrt(3)),N); MP("16",(8,0),S);MP("10",(18.5,5sqrt(3)/2),E);MP("4",(3,5sqrt(3)),N); dot((4,4sqrt(3))); MP("F",(4,4sqrt(3)),W); [/asy]

Let $ABCD$ be a parallelogram with $\angle{ABC}=120^\circ, AB=16$ and $BC=10.$ Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4.$ If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to

$\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution

Problem 5

Which of these numbers is largest?

$\textbf{(A) } \sqrt{\sqrt[3]{5\cdot 6}}\qquad \textbf{(B) } \sqrt{6\sqrt[3]{5}}\qquad \textbf{(C) } \sqrt{5\sqrt[3]{6}}\qquad \textbf{(D) } \sqrt[3]{5\sqrt{6}}\qquad \textbf{(E) } \sqrt[3]{6\sqrt{5}}$

Solution

Problem 6

Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$?

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

Solution

Problem 7

A triangle with integral sides has perimeter $8$. The area of the triangle is

$\textbf{(A) } 2\sqrt{2}\qquad \textbf{(B) } \frac{16}{9}\sqrt{3}\qquad \textbf{(C) } 2\sqrt{3} \qquad \textbf{(D) } 4\qquad \textbf{(E) } 4\sqrt{2}$

Solution

Problem 8

The number of real solutions of the equation \[|x-2|+|x-3|=1\] is

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

Solution

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

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

Solution

Problem 10

An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

$\textbf{(A) } 328\qquad \textbf{(B) } 329\qquad \textbf{(C) } 330\qquad \textbf{(D) } 331\qquad \textbf{(E) } 332$

Solution

Problem 11

How many positive integers less than $50$ have an odd number of positive integer divisors?

$\textbf{(A) } 3\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 9\qquad \textbf{(E) } 11$

Solution

Problem 12

Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$

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

Solution

Problem 13

If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$? 

1. START $X$ AT $3$ AND $S$ AT $0$.  
2. INCREASE THE VALUE OF $X$ BY $2$.   
3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$. 
4. IF $S$ IS AT LEAST $10000$,   
       THEN GO TO INSTRUCTION $5$;  
       OTHERWISE, GO TO INSTRUCTION $2$.  
       AND PROCEED FROM THERE.  
5. PRINT THE VALUE OF $X$.  
6.STOP.

$\textbf{(A) } 19\qquad \textbf{(B) } 21\qquad \textbf{(C) } 23\qquad \textbf{(D) } 199\qquad \textbf{(E) } 201$

Solution

Problem 14

[asy] draw(circle((0,0),1),black); draw((0,1)--(cos(pi/14),-sin(pi/14))--(-cos(pi/14),-sin(pi/14))--cycle,dot); draw((-cos(pi/14),-sin(pi/14))--(0,-1/cos(3pi/7))--(cos(pi/14),-sin(pi/14)),dot); draw(arc((0,1),.25,230,310)); MP("A",(0,1),N);MP("B",(cos(pi/14),-sin(pi/14)),E);MP("C",(-cos(pi/14),-sin(pi/14)),W);MP("D",(0,-1/cos(3pi/7)),S); MP("x",(0,.8),S); [/asy]

An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle{ABC}=\angle{ACB}=2\angle{D}$ and $x$ is the radian measure of $\angle{A}$, then $x=$

$\textbf{(A) } \frac{3\pi}{7}\qquad \textbf{(B) } \frac{4\pi}{9}\qquad \textbf{(C) } \frac{5\pi}{11}\qquad \textbf{(D) } \frac{6\pi}{13}\qquad \textbf{(E) } \frac{7\pi}{15}$

Solution

Problem 15

Four whole numbers, when added three at a time, give the sums $180,197,208$ and $222$. What is the largest of the four numbers?

$\textbf{(A) } 77\qquad \textbf{(B) } 83\qquad \textbf{(C) } 89\qquad \textbf{(D) } 95\qquad \textbf{(E) cannot be determined from the given information}$

Solution

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 $13$ married couples attended, how many handshakes were there among these $26$ people?

$\textbf{(A) } 78\qquad \textbf{(B) } 185\qquad \textbf{(C) } 234\qquad \textbf{(D) } 312\qquad \textbf{(E) } 325$

Solution

Problem 17

How many of the numbers, $100,101,\cdots,999$ have three different digits in increasing order or in decreasing order?

$\textbf{(A) } 120\qquad \textbf{(B) } 168\qquad \textbf{(C) } 204\qquad \textbf{(D) } 216\qquad \textbf{(E) } 240$

Solution

Problem 18

First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is

$\textbf{(A) } \frac{1}{16}\qquad \textbf{(B) } \frac{1}{8}\qquad \textbf{(C) } \frac{3}{16}\qquad \textbf{(D) } \frac{1}{5}\qquad \textbf{(E) } \frac{1}{4}$

Solution

Problem 19

For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ $\underline{not}$ in lowest terms?

$\textbf{(A) } 0\qquad \textbf{(B) } 86\qquad \textbf{(C) } 90\qquad \textbf{(D) } 104\qquad \textbf{(E) } 105$

Solution

Problem 20

[asy] draw((0,0)--(7,4.2)--(10,0)--(3,-5)--cycle,dot); draw((0,0)--(3,0)--(7,0)--(10,0),dot); draw((3,-5)--(3,0)--(7,0)--(7,4.2),dot); draw((3/sqrt(34),-5/sqrt(34))--(3/sqrt(34)+1/sqrt(1.36),-5/sqrt(34)+.6/sqrt(1.36))--(1/sqrt(1.36),.6/sqrt(1.36)),black+linewidth(.5)); draw((10-7/sqrt(74),0-5/sqrt(74))--(10-7/sqrt(74)-5/sqrt(74),0-5/sqrt(74)+7/sqrt(74))--(10-5/sqrt(74),7/sqrt(74)),black+linewidth(.5)); draw((3,-1)--(4,-1)--(4,0),black+linewidth(.5)); draw((6,0)--(6,1)--(7,1),black+linewidth(.5)); MP("A",(0,0),W);MP("B",(7,4.2),N);MP("C",(10,0),E);MP("D",(3,-5),S);MP("E",(3,0),N);MP("F",(7,0),S); [/asy]

In the figure $ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $\overline{AC}$, and $\overline{DE}$ and $\overline{BF}$ are perpendicular to $\overline{AC}$. If $AE=3, DE=5,$ and $CE=7$, then $BF=$

$\textbf{(A) } 3.6\qquad \textbf{(B) } 4\qquad \textbf{(C) } 4.2\qquad \textbf{(D) } 4.5\qquad \textbf{(E) } 5$

Solution

Problem 21

Consider a pyramid $P-ABCD$ whose base $ABCD$ is square and whose vertex $P$ is equidistant from $A,B,C$ and $D$. If $AB=1$ and $\angle{APB}=2\theta$, then the volume of the pyramid is

$\textbf{(A) } \frac{\sin(\theta)}{6}\qquad \textbf{(B) } \frac{\cot(\theta)}{6}\qquad \textbf{(C) } \frac{1}{6\sin(\theta)}\qquad \textbf{(D) } \frac{1-\sin(2\theta)}{6}\qquad \textbf{(E) } \frac{\sqrt{\cos(2\theta)}}{6\sin(\theta)}$

Solution

Problem 22

If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, then the product of those solutions with $a>0$ is

$\textbf{(A) } -2\qquad \textbf{(B) } 0\qquad \textbf{(C) } 2i\qquad \textbf{(D) } 4\qquad \textbf{(E) } 16$

Solution

Problem 23

If $x,y>0, \log_y(x)+\log_x(y)=\frac{10}{3} \text{ and } xy=144,\text{ then }\frac{x+y}{2}=$

$\textbf{(A) } 12\sqrt{2}\qquad \textbf{(B) } 13\sqrt{3}\qquad \textbf{(C) } 24\qquad \textbf{(D) } 30\qquad \textbf{(E) } 36$

Solution

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? \[\begin{tabular}[t]{|c|c|c|c|} \multicolumn{4}{c}{Average Scores}\\\hline Category&Adams&Baker&Adams\&Baker\\\hline Boys&71&81&79\\ Girls&76&90&?\\ Boys\&Girls&74&84& \\\hline \end{tabular}\] $\textbf{(A) } 81\qquad \textbf{(B) } 82\qquad \textbf{(C) } 83\qquad \textbf{(D) } 84\qquad \textbf{(E) } 85$

Solution

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?

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

Solution

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.) [asy] unitsize(2 cm); for(int i = 1; i <= 10; ++i) { label("``" + (string) i + "&#039;&#039;", dir(90 - 360/10*(i - 1))); } [/asy] The number picked by the person who announced the average $6$ was

$\textbf{(A) } 1 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 10 \qquad \textbf{(E) }\text{not uniquely determined from the given information}$

Solution

Problem 27

Which of these triples could $\underline{not}$ be the lengths of the three altitudes of a triangle?

$\textbf{(A) } 1,\sqrt{3},2\qquad \textbf{(B) } 3,4,5\qquad \textbf{(C) } 5,12,13\qquad \textbf{(D) } 7,8,\sqrt{113}\qquad \textbf{(E) } 8,15,17$

Solution

Problem 28

A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ 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 $x$ and $y$. Find $|x-y|$.

$\textbf{(A) } 12\qquad \textbf{(B) } 13\qquad \textbf{(C) } 14\qquad \textbf{(D) } 15\qquad \textbf{(E) } 16$

Solution

Problem 29

A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

$\textbf{(A) } 50\qquad \textbf{(B) } 66\qquad \textbf{(C) } 67\qquad \textbf{(D) } 76\qquad \textbf{(E) } 78$

Solution

Problem 30

If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, and $n=0,1,2,\cdots,$ then $R_{12345}$ is an integer. Its units digit is

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

Solution


See also

1990 AHSME (Problems • Answer Key • Resources)
Preceded by
1989 AHSME 		Followed by
1991 AHSME
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions


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