https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&feed=atom&action=history
1963 AHSME Problems - Revision history
2024-03-28T15:05:29Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=118239&oldid=prev
Made in 2016: /* See also */
2020-02-20T18:17:22Z
<p><span dir="auto"><span class="autocomment">See also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:17, 20 February 2020</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME 40p box|year=1963|before=[[1962 AHSME]]|after=[[1964 AHSME]]}}   </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME 40p box|year=1963|before=[[1962 AHSME<ins class="diffchange diffchange-inline">|1962 AHSC</ins>]]|after=[[1964 AHSME<ins class="diffchange diffchange-inline">|1964 AHSC</ins>]]}}   </div></td></tr>
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Made in 2016
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=118072&oldid=prev
Made in 2016 at 18:35, 19 February 2020
2020-02-19T18:35:17Z
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{AHSC 40 Problems</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|year = 1963</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 1==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 1==</div></td></tr>
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Made in 2016
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=94962&oldid=prev
Rockmanex3 at 06:50, 7 June 2018
2018-06-07T06:50:15Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:50, 7 June 2018</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME box|year=1963|before=[[1962 AHSME]]|after=[[1964 AHSME]]}}   </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME <ins class="diffchange diffchange-inline">40p </ins>box|year=1963|before=[[1962 AHSME]]|after=[[1964 AHSME]]}}   </div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=94926&oldid=prev
Rockmanex3: Fixing numbering error
2018-06-05T12:20:34Z
<p>Fixing numbering error</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 12:20, 5 June 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l126" >Line 126:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 10|Solution]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 10|Solution]]</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== Problem <del class="diffchange diffchange-inline">12</del>==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== Problem <ins class="diffchange diffchange-inline">11</ins>==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The arithmetic mean of a set of <math>50</math> numbers is <math>38</math>. If two numbers of the set, namely <math>45</math> and <math>55</math>, are discarded,  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The arithmetic mean of a set of <math>50</math> numbers is <math>38</math>. If two numbers of the set, namely <math>45</math> and <math>55</math>, are discarded,  </div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=68900&oldid=prev
Mathgeek2006 at 01:48, 14 March 2015
2015-03-14T01:48:50Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 01:48, 14 March 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >Line 3:</td>
<td colspan="2" class="diff-lineno">Line 3:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>?</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\(0,0)\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ (0,0)\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ \left(-\frac{1}{2},-1\right)\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ \left(-\frac{1}{2},-1\right)\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \left(\frac{1}{2},\frac{1}{3}\right)\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \left(\frac{1}{2},\frac{1}{3}\right)\qquad</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 36==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 36==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A person starting with <math>\<del class="diffchange diffchange-inline">textdollar </del>64</math> and making <math>6</math> bets, wins three times and loses three times,  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A person starting with <math>\<ins class="diffchange diffchange-inline">$</ins>64</math> and making <math>6</math> bets, wins three times and loses three times,  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss.  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\<del class="diffchange diffchange-inline">text</del>{(A) a loss of }\<del class="diffchange diffchange-inline">textdollar </del>27<del class="diffchange diffchange-inline">} </del>\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\<ins class="diffchange diffchange-inline">textbf</ins>{(A)<ins class="diffchange diffchange-inline">}\text{ </ins>a loss of }\<ins class="diffchange diffchange-inline">$ </ins>27 \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">text</del>{(B) a gain of }\<del class="diffchange diffchange-inline">textdollar </del>27<del class="diffchange diffchange-inline">} </del>\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">textbf</ins>{(B)<ins class="diffchange diffchange-inline">}\text{ </ins>a gain of }\<ins class="diffchange diffchange-inline">$ </ins>27 \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">text</del>{(C) a loss of }\<del class="diffchange diffchange-inline">textdollar </del>37<del class="diffchange diffchange-inline">} </del>\qquad \\</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">textbf</ins>{(C)<ins class="diffchange diffchange-inline">}\text{ </ins>a loss of }\<ins class="diffchange diffchange-inline">$ </ins>37 \qquad \\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">text</del>{(D) neither a gain nor a loss}\qquad \\</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">textbf</ins>{(D)<ins class="diffchange diffchange-inline">}\text{ </ins>neither a gain nor a loss}\qquad \\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">text</del>{(E) a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">textbf</ins>{(E)<ins class="diffchange diffchange-inline">}\text{ </ins>a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Mathgeek2006
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=65631&oldid=prev
Timneh: /* See also */
2014-10-23T06:27:09Z
<p><span dir="auto"><span class="autocomment">See also</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 06:27, 23 October 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l598" >Line 598:</td>
<td colspan="2" class="diff-lineno">Line 598:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">* [[AHSME]]</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">* [[AHSME Problems and Solutions]]</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[AMC 12 Problems and Solutions]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[AMC 12 Problems and Solutions]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Mathematics competition resources]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Mathematics competition resources]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{AHSME box|year=1963|before=[[1962 AHSME]]|after=[[1964 AHSME]]}}  </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{MAA Notice}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{MAA Notice}}</div></td></tr>
</table>
Timneh
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=64831&oldid=prev
Timneh: /* Problem 36 */
2014-10-10T02:50:27Z
<p><span dir="auto"><span class="autocomment">Problem 36</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:50, 10 October 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l501" >Line 501:</td>
<td colspan="2" class="diff-lineno">Line 501:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\<del class="diffchange diffchange-inline">textbf</del>{(A) a loss of }\textdollar 27} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\<ins class="diffchange diffchange-inline">text</ins>{(A) a loss of }\textdollar 27} \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">textbf</del>{(B) a gain of }\textdollar 27} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">text</ins>{(B) a gain of }\textdollar 27} \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">textbf</del>{(C) a loss of }\textdollar 37} \qquad \\</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">text</ins>{(C) a loss of }\textdollar 37} \qquad \\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">textbf</del>{(D) neither a gain nor a loss}\qquad \\</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">text</ins>{(D) neither a gain nor a loss}\qquad \\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">textbf</del>{(E) a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">text</ins>{(E) a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Timneh
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=64830&oldid=prev
Timneh at 02:49, 10 October 2014
2014-10-10T02:49:58Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:49, 10 October 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l63" >Line 63:</td>
<td colspan="2" class="diff-lineno">Line 63:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\triangle BAD</math> is right-angled at <math>B</math>. On <math>AD</math> there is a point <math>C</math> for which <math>AC=CD</math> and <math>AB=BC</math>. The magnitude of <math>\angle DAB</math> is:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\triangle BAD</math> is right-angled at <math>B</math>. On <math>AD</math> there is a point <math>C</math> for which <math>AC=CD</math> and <math>AB=BC</math>. The magnitude of <math>\angle DAB</math> is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ 67\<del class="diffchange diffchange-inline">frac</del>{1}{2}^{\circ}\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ 67\<ins class="diffchange diffchange-inline">tfrac</ins>{1}{2}^{\circ}\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ 60^{\circ}\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ 60^{\circ}\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ 45^{\circ}\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ 45^{\circ}\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ 30^{\circ}\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ 30^{\circ}\qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ 22\<del class="diffchange diffchange-inline">frac</del>{1}{2}^{\circ} </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ 22\<ins class="diffchange diffchange-inline">tfrac</ins>{1}{2}^{\circ} </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 6|Solution]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 6|Solution]]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l217" >Line 217:</td>
<td colspan="2" class="diff-lineno">Line 217:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 18==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 18==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Chord <math>EF</math> is the perpendicular bisector of chord <math>BC</math>, intersecting it in <math>M</math>. Between <math>B</math> and <math>M</math> point <math>U</math> is taken,  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Chord <math>EF</math> is the perpendicular bisector of chord <math>BC</math>, intersecting it in <math>M</math>.  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>and <math><del class="diffchange diffchange-inline">EU4 </del>extended meets the circle in <<del class="diffchange diffchange-inline">/</del>math>A<math>. Then, for any selection of <<del class="diffchange diffchange-inline">/</del>math>U<math>, as described, <<del class="diffchange diffchange-inline">/</del>math>\triangle EUM<del class="diffchange diffchange-inline">$ </del>is similar to <del class="diffchange diffchange-inline">triangle</del>:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Between <math>B</math> and <math>M</math> point <math>U</math> is taken,  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>and <math><ins class="diffchange diffchange-inline">EU</math> </ins>extended meets the circle in <math>A<<ins class="diffchange diffchange-inline">/</ins>math>. Then, for any selection of <math>U<<ins class="diffchange diffchange-inline">/</ins>math>, as described, <math>\triangle EUM<ins class="diffchange diffchange-inline"></math> </ins>is similar to:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><asy></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><asy></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l241" >Line 241:</td>
<td colspan="2" class="diff-lineno">Line 242:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ EFA \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ <ins class="diffchange diffchange-inline">\triangle </ins>EFA \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ EFC \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ <ins class="diffchange diffchange-inline">\triangle </ins>EFC \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ ABM \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ <ins class="diffchange diffchange-inline">\triangle </ins>ABM \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ ABU \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ <ins class="diffchange diffchange-inline">\triangle </ins>ABU \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ FMC    </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ <ins class="diffchange diffchange-inline">\triangle </ins>FMC    </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 18|Solution]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 18|Solution]]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l281" >Line 281:</td>
<td colspan="2" class="diff-lineno">Line 282:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The expression <math>x^2-y^2-z^2+2yz+x+y-z</math> has:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The expression <math>x^2-y^2-z^2+2yz+x+y-z</math> has:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ \text{no linear factor with integer coefficients and integer exponents} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ \text{no linear factor with integer coefficients and integer exponents} \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ \text{the factor }-x+y+z \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ \text{the factor }-x+y+z \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ \text{the factor }x-y+z+1    </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ \text{the factor }x-y+z+1    </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l462" >Line 462:</td>
<td colspan="2" class="diff-lineno">Line 463:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ y =\frac{3}{4}x+1\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ y =\frac{3}{4}x+1\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ y =\frac{3}{4}x\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ y =\frac{3}{4}x\qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ y =\frac{3}{4}x-\frac{2}{3}\qquad  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ y =\frac{3}{4}x-\frac{2}{3}\qquad <ins class="diffchange diffchange-inline"> \\</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ y = \dfrac{3}{4}x + 2    </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ y = \dfrac{3}{4}x + 2    </math></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l496" >Line 496:</td>
<td colspan="2" class="diff-lineno">Line 497:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 36==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Problem 36==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A person starting with <del class="diffchange diffchange-inline">&#036;</del>64 and making 6 bets, wins three times and loses three times,  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A person starting with <ins class="diffchange diffchange-inline"><math>\textdollar </ins>64<ins class="diffchange diffchange-inline"></math> </ins>and making <ins class="diffchange diffchange-inline"><math></ins>6<ins class="diffchange diffchange-inline"></math> </ins>bets, wins three times and loses three times,  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss.  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If each wager is for half the money remaining at the time of the bet, then the final result is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)<del class="diffchange diffchange-inline">}\ \text{</del>a loss of <del class="diffchange diffchange-inline">&#036;</del>27} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A) a loss of <ins class="diffchange diffchange-inline">}\textdollar </ins>27} \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)<del class="diffchange diffchange-inline">}\ \text{</del>a gain of <del class="diffchange diffchange-inline">&#036;</del>27} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B) a gain of <ins class="diffchange diffchange-inline">}\textdollar </ins>27} \qquad</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)<del class="diffchange diffchange-inline">}\ \text{</del>a loss of <del class="diffchange diffchange-inline">&#036;</del>37} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C) a loss of <ins class="diffchange diffchange-inline">}\textdollar </ins>37} \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)<del class="diffchange diffchange-inline">}\ \text{</del>neither a gain nor a loss}\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D) neither a gain nor a loss}\qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)<del class="diffchange diffchange-inline">}\ \text{</del>a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E) a gain or a loss depending upon the order in which the wins and losses occur} </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l594" >Line 594:</td>
<td colspan="2" class="diff-lineno">Line 595:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ 95\text{ and }105 </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ 95\text{ and }105 </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> </del>[[1963 AHSME Problems/Problem 40|Solution]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[1963 AHSME Problems/Problem 40|Solution]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
</table>
Timneh
https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems&diff=64829&oldid=prev
Timneh: Created page with "== Problem 1== Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>? <math>\textbf{(A)}\(0,0)\qquad \textbf{(B)}\ \left(-\frac{1}{2},-1\right)..."
2014-10-10T02:43:02Z
<p>Created page with "== Problem 1== Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>? <math>\textbf{(A)}\(0,0)\qquad \textbf{(B)}\ \left(-\frac{1}{2},-1\right)..."</p>
<p><b>New page</b></p><div>== Problem 1==<br />
<br />
Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>?<br />
<br />
<math>\textbf{(A)}\(0,0)\qquad<br />
\textbf{(B)}\ \left(-\frac{1}{2},-1\right)\qquad<br />
\textbf{(C)}\ \left(\frac{1}{2},\frac{1}{3}\right)\qquad<br />
\textbf{(D)}\ (-1,1)\qquad<br />
\textbf{(E)}\ (-2,2) </math> <br />
<br />
[[1963 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2==<br />
<br />
let <math>n=x-y^{x-y}</math>. Find <math>n</math> when <math>x=2</math> and <math>y=-2</math>.<br />
<br />
<math>\textbf{(A)}\ -14 \qquad<br />
\textbf{(B)}\ 0 \qquad<br />
\textbf{(C)}\ 1 \qquad<br />
\textbf{(D)}\ 18 \qquad<br />
\textbf{(E)}\ 256 </math><br />
<br />
[[1963 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3==<br />
<br />
If the reciprocal of <math>x+1</math> is <math>x-1</math>, then <math>x</math> equals:<br />
<br />
<math>\textbf{(A)}\ 0\qquad<br />
\textbf{(B)}\ 1\qquad<br />
\textbf{(C)}\ -1\qquad<br />
\textbf{(D)}\ \pm 1\qquad<br />
\textbf{(E)}\ \text{none of these} </math> <br />
<br />
[[1963 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4==<br />
<br />
For what value(s) of <math>k</math> does the pair of equations <math>y=x^2</math> and <math>y=3x+k</math> have two identical solutions?<br />
<br />
<math>\textbf{(A)}\ \frac{4}{9}\qquad<br />
\textbf{(B)}\ -\frac{4}{9}\qquad<br />
\textbf{(C)}\ \frac{9}{4}\qquad<br />
\textbf{(D)}\ -\frac{9}{4}\qquad<br />
\textbf{(E)}\ \pm\frac{9}{4} </math> <br />
<br />
[[1963 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5==<br />
<br />
If <math>x</math> and <math>\log_{10} x</math> are real numbers and <math>\log_{10} x<0</math>, then:<br />
<br />
<math>\textbf{(A)}\ x<0 \qquad<br />
\textbf{(B)}\ -1<x<1 \qquad<br />
\textbf{(C)}\ 0<x\le 1 \\<br />
\textbf{(D)}\ -1<x<0 \qquad<br />
\textbf{(E)}\ 0<x<1 </math><br />
<br />
[[1963 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6==<br />
<br />
<math>\triangle BAD</math> is right-angled at <math>B</math>. On <math>AD</math> there is a point <math>C</math> for which <math>AC=CD</math> and <math>AB=BC</math>. The magnitude of <math>\angle DAB</math> is:<br />
<br />
<math>\textbf{(A)}\ 67\frac{1}{2}^{\circ}\qquad<br />
\textbf{(B)}\ 60^{\circ}\qquad<br />
\textbf{(C)}\ 45^{\circ}\qquad<br />
\textbf{(D)}\ 30^{\circ}\qquad<br />
\textbf{(E)}\ 22\frac{1}{2}^{\circ} </math><br />
<br />
[[1963 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7==<br />
<br />
Given the four equations:<br />
<br />
<math>\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10</math><br />
<br />
The pair representing the perpendicular lines is:<br />
<br />
<br />
<math>\textbf{(A)}\ \text{(1) and (4)}\qquad<br />
\textbf{(B)}\ \text{(1) and (3)}\qquad<br />
\textbf{(C)}\ \text{(1) and (2)}\qquad<br />
\textbf{(D)}\ \text{(2) and (4)}\qquad<br />
\textbf{(E)}\ \text{(2) and (3)} </math> <br />
<br />
[[1963 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8==<br />
<br />
The smallest positive integer <math>x</math> for which <math>1260x=N^3</math>, where <math>N</math> is an integer, is:<br />
<br />
<br />
<math>\textbf{(A)}\ 1050 \qquad<br />
\textbf{(B)}\ 1260 \qquad<br />
\textbf{(C)}\ 1260^2 \qquad<br />
\textbf{(D)}\ 7350 \qquad<br />
\textbf{(E)}\ 44100 </math><br />
<br />
[[1963 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9==<br />
<br />
In the expansion of <math>\left(a-\dfrac{1}{\sqrt{a}}\right)^7</math> the coefficient of <math>a^{-\dfrac{1}{2}}</math> is:<br />
<br />
<math>\textbf{(A)}\ -7 \qquad<br />
\textbf{(B)}\ 7 \qquad<br />
\textbf{(C)}\ -21 \qquad<br />
\textbf{(D)}\ 21 \qquad<br />
\textbf{(E)}\ 35 </math><br />
<br />
[[1963 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10==<br />
<br />
Point <math>P</math> is taken interior to a square with side-length <math>a</math> and such that is it equally distant from two <br />
consecutive vertices and from the side opposite these vertices. If <math>d</math> represents the common distance, then <math>d</math> equals:<br />
<br />
<math>\textbf{(A)}\ \frac{3a}{5}\qquad<br />
\textbf{(B)}\ \frac{5a}{8}\qquad<br />
\textbf{(C)}\ \frac{3a}{8}\qquad<br />
\textbf{(D)}\ \frac{a\sqrt{2}}{2}\qquad<br />
\textbf{(E)}\ \frac{a}{2} </math> <br />
<br />
[[1963 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 12==<br />
<br />
The arithmetic mean of a set of <math>50</math> numbers is <math>38</math>. If two numbers of the set, namely <math>45</math> and <math>55</math>, are discarded, <br />
the arithmetic mean of the remaining set of numbers is:<br />
<br />
<math>\textbf{(A)}\ 38.5 \qquad<br />
\textbf{(B)}\ 37.5 \qquad<br />
\textbf{(C)}\ 37 \qquad<br />
\textbf{(D)}\ 36.5 \qquad<br />
\textbf{(E)}\ 36 </math><br />
<br />
[[1963 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12==<br />
<br />
Three vertices of parallelogram <math>PQRS</math> are <math>P(-3,-2), Q(1,-5), R(9,1)</math> with <math>P</math> and <math>R</math> diagonally opposite. <br />
The sum of the coordinates of vertex <math>S</math> is:<br />
<br />
<math>\textbf{(A)}\ 13 \qquad<br />
\textbf{(B)}\ 12 \qquad<br />
\textbf{(C)}\ 11 \qquad<br />
\textbf{(D)}\ 10 \qquad<br />
\textbf{(E)}\ 9 </math><br />
<br />
[[1963 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13==<br />
<br />
If <math>2^a+2^b=3^c+3^d</math>, the number of integers <math>a,b,c,d</math> which can possibly be negative, is, at most:<br />
<br />
<math>\textbf{(A)}\ 4 \qquad<br />
\textbf{(B)}\ 3 \qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ 1 \qquad<br />
\textbf{(E)}\ 0 </math><br />
<br />
[[1963 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14==<br />
<br />
Given the equations <math>x^2+kx+6=0</math> and <math>x^2-kx+6=0</math>. If, when the roots of the equation are suitably listed, <br />
each root of the second equation is <math>5</math> more than the corresponding root of the first equation, then <math>k</math> equals:<br />
<br />
<math>\textbf{(A)}\ 5 \qquad<br />
\textbf{(B)}\ -5 \qquad<br />
\textbf{(C)}\ 7 \qquad<br />
\textbf{(D)}\ -7 \qquad<br />
\textbf{(E)}\ \text{none of these} </math><br />
<br />
[[1963 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15==<br />
<br />
A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. <br />
The ratio of the area of the triangle to the area of the square is:<br />
<br />
<math>\textbf{(A)}\ \sqrt{3}:1\qquad<br />
\textbf{(B)}\ \sqrt{3}:\sqrt{2}\qquad<br />
\textbf{(C)}\ 3\sqrt{3}:2\qquad<br />
\textbf{(D)}\ 3:\sqrt{2}\qquad<br />
\textbf{(E)}\ 3:2\sqrt{2} </math><br />
<br />
[[1963 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16==<br />
<br />
Three numbers <math>a,b,c</math>, none zero, form an arithmetic progression. Increasing <math>a</math> by <math>1</math> or increasing <math>c</math> by <math>2</math> results <br />
in a geometric progression. Then <math>b</math> equals:<br />
<br />
<math>\textbf{(A)}\ 16 \qquad<br />
\textbf{(B)}\ 14 \qquad<br />
\textbf{(C)}\ 12 \qquad<br />
\textbf{(D)}\ 10 \qquad<br />
\textbf{(E)}\ 8 </math><br />
<br />
[[1963 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17==<br />
<br />
The expression <math>\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}</math>, <math>a</math> real, <math>a\neq 0</math>, has the value <math>-1</math> for:<br />
<br />
<math>\textbf{(A)}\ \text{all but two real values of }y \qquad \\<br />
\textbf{(B)}\ \text{only two real values of }y \qquad \\<br />
\textbf{(C)}\ \text{all real values of }y\qquad \\<br />
\textbf{(D)}\ \text{only one real value of }y\qquad \\<br />
\textbf{(E)}\ \text{no real values of }y </math> <br />
<br />
[[1963 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18==<br />
<br />
Chord <math>EF</math> is the perpendicular bisector of chord <math>BC</math>, intersecting it in <math>M</math>. Between <math>B</math> and <math>M</math> point <math>U</math> is taken, <br />
and <math>EU4 extended meets the circle in </math>A<math>. Then, for any selection of </math>U<math>, as described, </math>\triangle EUM$ is similar to triangle:<br />
<br />
<asy><br />
pair B = (-0.866, -0.5);<br />
pair C = (0.866, -0.5);<br />
pair E = (0, -1);<br />
pair F = (0, 1);<br />
pair M = midpoint(B--C);<br />
pair A = (-0.99, -0.141);<br />
pair U = intersectionpoints(A--E, B--C)[0];<br />
draw(B--C);<br />
draw(F--E--A);<br />
draw(unitcircle);<br />
label("$B$", B, SW);<br />
label("$C$", C, SE);<br />
label("$A$", A, W);<br />
label("$E$", E, S);<br />
label("$U$", U, NE);<br />
label("$M$", M, NE);<br />
label("$F$", F, N);<br />
//Credit to MSTang for the asymptote</asy><br />
<br />
<br />
<math>\textbf{(A)}\ EFA \qquad<br />
\textbf{(B)}\ EFC \qquad<br />
\textbf{(C)}\ ABM \qquad<br />
\textbf{(D)}\ ABU \qquad<br />
\textbf{(E)}\ FMC </math><br />
<br />
[[1963 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19==<br />
<br />
In counting <math>n</math> colored balls, some red and some black, it was found that <math>49</math> of the first <math>50</math> counted were red. <br />
Thereafter, <math>7</math> out of every <math>8</math> counted were red. If, in all, <math>90</math> % or more of the balls counted were red, the maximum value of <math>n</math> is:<br />
<br />
<math>\textbf{(A)}\ 225 \qquad<br />
\textbf{(B)}\ 210 \qquad<br />
\textbf{(C)}\ 200 \qquad<br />
\textbf{(D)}\ 180 \qquad<br />
\textbf{(E)}\ 175 </math><br />
<br />
[[1963 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20==<br />
<br />
Two men at points <math>R</math> and <math>S</math>, <math>76</math> miles apart, set out at the same time to walk towards each other. <br />
The man at <math>R</math> walks uniformly at the rate of <math>4\tfrac{1}{2}</math> miles per hour; the man at <math>S</math> walks at the constant <br />
rate of <math>3\tfrac{1}{4}</math> miles per hour for the first hour, at <math>3\tfrac{3}{4}</math> miles per hour for the second hour,<br />
and so on, in arithmetic progression. If the men meet <math>x</math> miles nearer <math>R</math> than <math>S</math> in an integral number of hours, then <math>x</math> is:<br />
<br />
<math>\textbf{(A)}\ 10 \qquad<br />
\textbf{(B)}\ 8 \qquad<br />
\textbf{(C)}\ 6 \qquad<br />
\textbf{(D)}\ 4 \qquad<br />
\textbf{(E)}\ 2 </math><br />
<br />
[[1963 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21==<br />
<br />
The expression <math>x^2-y^2-z^2+2yz+x+y-z</math> has:<br />
<br />
<math>\textbf{(A)}\ \text{no linear factor with integer coefficients and integer exponents} \qquad<br />
\textbf{(B)}\ \text{the factor }-x+y+z \qquad<br />
\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad<br />
\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad<br />
\textbf{(E)}\ \text{the factor }x-y+z+1 </math><br />
<br />
[[1963 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22==<br />
<br />
Acute-angled <math>\triangle ABC</math> is inscribed in a circle with center at <math>O</math>; <math>\stackrel \frown {AB} = 120</math> and <math>\stackrel \frown {BC} = 72</math>. <br />
<math>A</math> point <math>E</math> is taken in minor arc <math>AC</math> such that <math>OE</math> is perpendicular to <math>AC</math>. Then the ratio of the magnitudes of <math>\angle OBE</math> and <math>\angle BAC</math> is:<br />
<br />
<math>\textbf{(A)}\ \frac{5}{18}\qquad<br />
\textbf{(B)}\ \frac{2}{9}\qquad<br />
\textbf{(C)}\ \frac{1}{4}\qquad<br />
\textbf{(D)}\ \frac{1}{3}\qquad<br />
\textbf{(E)}\ \frac{4}{9} </math><br />
<br />
[[1963 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23==<br />
<br />
A gives <math>B</math> as many cents as <math>B</math> has and <math>C</math> as many cents as <math>C</math> has. Similarly, <math>B</math> then gives <math>A</math> and <math>C</math> as many cents as each then has. <br />
<math>C</math>, similarly, then gives <math>A</math> and <math>B</math> as many cents as each then has. If each finally has <math>16</math> cents, with how many cents does <math>A</math> start?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad<br />
\textbf{(B)}\ 26\qquad<br />
\textbf{(C)}\ 28 \qquad<br />
\textbf{(D)}\ 30 \qquad<br />
\textbf{(E)}\ 32 </math><br />
<br />
[[1963 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24==<br />
<br />
Consider equations of the form <math>x^2 + bx + c = 0</math>. How many such equations have real roots and have coefficients <math>b</math> and <math>c</math> selected <br />
from the set of integers <math>\{1,2,3, 4, 5,6\}</math>?<br />
<br />
<math>\textbf{(A)}\ 20 \qquad<br />
\textbf{(B)}\ 19 \qquad<br />
\textbf{(C)}\ 18 \qquad<br />
\textbf{(D)}\ 17 \qquad<br />
\textbf{(E)}\ 16 </math><br />
<br />
[[1963 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25==<br />
<br />
Point <math>F</math> is taken in side <math>AD</math> of square <math>ABCD</math>. At <math>C</math> a perpendicular is drawn to <math>CF</math>, meeting <math>AB</math> extended at <math>E</math>. <br />
The area of <math>ABCD</math> is <math>256</math> square inches and the area of <math>\triangle CEF</math> is <math>200</math> square inches. Then the number of inches in <math>BE</math> is:<br />
<br />
<asy><br />
size(6cm);<br />
pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7);<br />
draw(A--B--C--D--cycle);<br />
draw(F--C--E--B);<br />
label("$A$", A, SW);<br />
label("$B$", B, S);<br />
label("$C$", C, N);<br />
label("$D$", D, NW);<br />
label("$E$", E, SE);<br />
label("$F$", F, W);<br />
//Credit to MSTang for the asymptote</asy><br />
<br />
<math>\textbf{(A)}\ 12 \qquad<br />
\textbf{(B)}\ 14 \qquad<br />
\textbf{(C)}\ 15 \qquad<br />
\textbf{(D)}\ 16 \qquad<br />
\textbf{(E)}\ 20 </math><br />
<br />
[[1963 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26==<br />
<br />
Version 1<br />
Consider the statements:<br />
<br />
<math> \textbf{(1)}\ p\text{ }\wedge\sim q\wedge r\qquad\textbf{(2)}\ \sim p\text{ }\wedge\sim q\wedge r\qquad\textbf{(3)}\ p\text{ }\wedge\sim q\text{ }\wedge\sim r\qquad\textbf{(4)}\ \sim p\text{ }\wedge q\text{ }\wedge r </math><br />
<br />
where <math>p,q</math>, and <math>r</math> are propositions. How many of these imply the truth of <math>(p\rightarrow q)\rightarrow r</math>?<br />
<br />
<math>\textbf{(A)}\ 0 \qquad<br />
\textbf{(B)}\ 1\qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ 3 \qquad<br />
\textbf{(E)}\ 4</math><br />
<br />
Version 2<br />
Consider the statements (1) <math>p</math> and <math>r</math> are true and <math>q</math> is false (2) <math>r</math> is true and <math>p</math> and <math>q</math> are false (3) <math>p</math> is true and <math>q</math> and <math>r</math> are false (4) <math>q</math> and <math>r</math> are true and <math>p</math> is false. <br />
How many of these imply the truth of the statement<br />
"<math>r</math> is implied by the statement that <math>p</math> implies <math>q</math>"?<br />
<br />
<math>\textbf{(A)}\ 0 \qquad<br />
\textbf{(B)}\ 1\qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ 3 \qquad<br />
\textbf{(E)}\ 4 </math><br />
<br />
[[1963 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27==<br />
<br />
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:<br />
<br />
<math>\textbf{(A)}\ 16 \qquad<br />
\textbf{(B)}\ 20\qquad<br />
\textbf{(C)}\ 22 \qquad<br />
\textbf{(D)}\ 24 \qquad<br />
\textbf{(E)}\ 26 </math><br />
<br />
[[1963 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28==<br />
<br />
Given the equation <math>3x^2 - 4x + k = 0</math> with real roots. The value of <math>k</math> for which the product of the roots of the equation is a maximum is:<br />
<br />
<math>\textbf{(A)}\ \frac{16}{9}\qquad<br />
\textbf{(B)}\ \frac{16}{3}\qquad<br />
\textbf{(C)}\ \frac{4}{9}\qquad<br />
\textbf{(D)}\ \frac{4}{3}\qquad<br />
\textbf{(E)}\ -\frac{4}{3} </math> <br />
<br />
[[1963 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29==<br />
<br />
A particle projected vertically upward reaches, at the end of <math>t</math> seconds, an elevation of <math>s</math> feet where <math>s = 160 t - 16t^2</math>. The highest elevation is:<br />
<br />
<math>\textbf{(A)}\ 800 \qquad<br />
\textbf{(B)}\ 640\qquad<br />
\textbf{(C)}\ 400 \qquad<br />
\textbf{(D)}\ 320 \qquad<br />
\textbf{(E)}\ 160 </math><br />
<br />
[[1963 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30==<br />
<br />
Let <math>F=\log\dfrac{1+x}{1-x}</math>. Find a new function <math>G</math> by replacing each <math>x</math> in <math>F</math> by <math>\dfrac{3x+x^3}{1+3x^2}</math>, and simplify. <br />
The simplified expression <math>G</math> is equal to:<br />
<br />
<math>\textbf{(A)}\ -F \qquad<br />
\textbf{(B)}\ F\qquad<br />
\textbf{(C)}\ 3F \qquad<br />
\textbf{(D)}\ F^3 \qquad<br />
\textbf{(E)}\ F^3-F </math><br />
<br />
[[1963 AHSME Problems/Problem 30|Solution]]<br />
<br />
== Problem 31==<br />
<br />
The number of solutions in positive integers of <math>2x+3y=763</math> is:<br />
<br />
<math>\textbf{(A)}\ 255 \qquad<br />
\textbf{(B)}\ 254\qquad<br />
\textbf{(C)}\ 128 \qquad<br />
\textbf{(D)}\ 127 \qquad<br />
\textbf{(E)}\ 0 </math><br />
<br />
[[1963 AHSME Problems/Problem 31|Solution]]<br />
<br />
== Problem 32==<br />
<br />
The dimensions of a rectangle <math>R</math> are <math>a</math> and <math>b</math>, <math>a < b</math>. It is required to obtain a rectangle with dimensions <math>x</math> and <math>y</math>, <math>x < a, y < a</math>, <br />
so that its perimeter is one-third that of <math>R</math>, and its area is one-third that of <math>R</math>. The number of such (different) rectangles is:<br />
<br />
<math>\textbf{(A)}\ 0 \qquad<br />
\textbf{(B)}\ 1\qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ 4 \qquad<br />
\textbf{(E)}\ \infty </math><br />
<br />
[[1963 AHSME Problems/Problem 32|Solution]]<br />
<br />
== Problem 33==<br />
<br />
Given the line <math>y = \dfrac{3}{4}x + 6</math> and a line <math>L</math> parallel to the given line and <math>4</math> units from it. A possible equation for <math>L</math> is:<br />
<br />
<math>\textbf{(A)}\ y =\frac{3}{4}x+1\qquad<br />
\textbf{(B)}\ y =\frac{3}{4}x\qquad<br />
\textbf{(C)}\ y =\frac{3}{4}x-\frac{2}{3}\qquad <br />
\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad<br />
\textbf{(E)}\ y = \dfrac{3}{4}x + 2 </math><br />
<br />
[[1963 AHSME Problems/Problem 33|Solution]]<br />
<br />
== Problem 34==<br />
<br />
In <math>\triangle ABC</math>, side <math>a = \sqrt{3}</math>, side <math>b = \sqrt{3}</math>, and side <math>c > 3</math>. Let <math>x</math> be the largest number such that the magnitude, <br />
in degrees, of the angle opposite side <math>c</math> exceeds <math>x</math>. Then <math>x</math> equals:<br />
<br />
<math>\textbf{(A)}\ 150^{\circ} \qquad<br />
\textbf{(B)}\ 120^{\circ}\qquad<br />
\textbf{(C)}\ 105^{\circ} \qquad<br />
\textbf{(D)}\ 90^{\circ} \qquad<br />
\textbf{(E)}\ 60^{\circ} </math> <br />
<br />
[[1963 AHSME Problems/Problem 34|Solution]]<br />
<br />
== Problem 35==<br />
<br />
The lengths of the sides of a triangle are integers, and its area is also an integer. <br />
One side is <math>21</math> and the perimeter is <math>48</math>. The shortest side is:<br />
<br />
<math>\textbf{(A)}\ 8 \qquad<br />
\textbf{(B)}\ 10\qquad<br />
\textbf{(C)}\ 12 \qquad<br />
\textbf{(D)}\ 14 \qquad<br />
\textbf{(E)}\ 16 </math><br />
<br />
[[1963 AHSME Problems/Problem 35|Solution]]<br />
<br />
== Problem 36==<br />
<br />
A person starting with &#036;64 and making 6 bets, wins three times and loses three times, <br />
the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. <br />
If each wager is for half the money remaining at the time of the bet, then the final result is:<br />
<br />
<math>\textbf{(A)}\ \text{a loss of &#036;27} \qquad<br />
\textbf{(B)}\ \text{a gain of &#036;27} \qquad<br />
\textbf{(C)}\ \text{a loss of &#036;37} \qquad<br />
\textbf{(D)}\ \text{neither a gain nor a loss}\qquad<br />
\textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur} </math><br />
<br />
<br />
[[1963 AHSME Problems/Problem 36|Solution]]<br />
<br />
== Problem 37==<br />
<br />
Given points <math>P_1, P_2,\cdots,P_7</math> on a straight line, in the order stated (not necessarily evenly spaced). <br />
Let <math>P</math> be an arbitrarily selected point on the line and let <math>s</math> be the sum of the undirected lengths<br />
<math>PP_1, PP_2, \cdots , PP_7</math>. Then <math>s</math> is smallest if and only if the point <math>P</math> is:<br />
<br />
<math>\textbf{(A)}\ \text{midway between }P_1\text{ and }P_7\qquad \\<br />
\textbf{(B)}\ \text{midway between }P_2\text{ and }P_6\qquad \\<br />
\textbf{(C)}\ \text{midway between }P_3\text{ and }P_5\qquad \\<br />
\textbf{(D)}\ \text{at }P_4 \qquad<br />
\textbf{(E)}\ \text{at }P_1 </math><br />
<br />
[[1963 AHSME Problems/Problem 37|Solution]]<br />
<br />
== Problem 38==<br />
<br />
Point <math>F</math> is taken on the extension of side <math>AD</math> of parallelogram <math>ABCD</math>. <math>BF</math> intersects diagonal <math>AC</math> at <math>E</math> and side <math>DC</math> at <math>G</math>. <br />
If <math>EF = 32</math> and <math>GF = 24</math>, then <math>BE</math> equals:<br />
<br />
<asy><br />
size(7cm);<br />
pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5);<br />
pair G = intersectionpoints(B--F, D--C)[0];<br />
pair E = intersectionpoints(A--C, B--F)[0];<br />
draw(A--D--C--B--cycle);<br />
draw(A--C);<br />
draw(D--F--B);<br />
label("$A$", A, SW);<br />
label("$B$", B, SE);<br />
label("$C$", C, NE);<br />
label("$D$", D, NW);<br />
label("$F$", F, N);<br />
label("$G$", G, NE);<br />
label("$E$", E, SE);<br />
//Credit to MSTang for the asymptote</asy><br />
<br />
<math>\textbf{(A)}\ 4 \qquad<br />
\textbf{(B)}\ 8\qquad<br />
\textbf{(C)}\ 10 \qquad<br />
\textbf{(D)}\ 12 \qquad<br />
\textbf{(E)}\ 16 </math><br />
<br />
[[1963 AHSME Problems/Problem 38|Solution]]<br />
<br />
== Problem 39==<br />
<br />
In <math>\triangle ABC</math> lines <math>CE</math> and <math>AD</math> are drawn so that <br />
<math>\dfrac{CD}{DB}=\dfrac{3}{1}</math> and <math>\dfrac{AE}{EB}=\dfrac{3}{2}</math>. Let <math>r=\dfrac{CP}{PE}</math> <br />
where <math>P</math> is the intersection point of <math>CE</math> and <math>AD</math>. Then <math>r</math> equals:<br />
<br />
<asy><br />
size(8cm);<br />
pair A = (0, 0), B = (9, 0), C = (3, 6);<br />
pair D = (7.5, 1.5), E = (6.5, 0);<br />
pair P = intersectionpoints(A--D, C--E)[0];<br />
draw(A--B--C--cycle);<br />
draw(A--D);<br />
draw(C--E);<br />
label("$A$", A, SW);<br />
label("$B$", B, SE);<br />
label("$C$", C, N);<br />
label("$D$", D, NE);<br />
label("$E$", E, S);<br />
label("$P$", P, S);<br />
//Credit to MSTang for the asymptote</asy><br />
<br />
<br />
<math>\textbf{(A)}\ 3 \qquad<br />
\textbf{(B)}\ \dfrac{3}{2}\qquad<br />
\textbf{(C)}\ 4 \qquad<br />
\textbf{(D)}\ 5 \qquad<br />
\textbf{(E)}\ \dfrac{5}{2} </math> <br />
<br />
[[1963 AHSME Problems/Problem 39|Solution]]<br />
<br />
== Problem 40==<br />
<br />
If <math>x</math> is a number satisfying the equation <math>\sqrt[3]{x+9}-\sqrt[3]{x-9}=3</math>, then <math>x^2</math> is between:<br />
<br />
<math>\textbf{(A)}\ 55\text{ and }65\qquad<br />
\textbf{(B)}\ 65\text{ and }75\qquad<br />
\textbf{(C)}\ 75\text{ and }85\qquad<br />
\textbf{(D)}\ 85\text{ and }95\qquad<br />
\textbf{(E)}\ 95\text{ and }105 </math><br />
<br />
[[1963 AHSME Problems/Problem 40|Solution]]<br />
<br />
== See also ==<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
{{MAA Notice}}</div>
Timneh