https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&feed=atom&action=history2016 AIME I Problems/Problem 14 - Revision history2024-03-28T16:15:48ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=182865&oldid=prevPuffer13: /* Solution 3 (risky) */2022-11-23T01:08:47Z<p><span dir="auto"><span class="autocomment">Solution 3 (risky)</span></span></p>
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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>-Patrick4President</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>-Patrick4President</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 style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">==Solution 3 (risky)==</del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</del></div></td><td colspan="2"> </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;"></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>==Video 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>==Video Solution==</div></td></tr>
</table>Puffer13https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=173976&oldid=prevDaijobu at 22:30, 15 May 20222022-05-15T22:30:10Z<p></p>
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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>==Solution 3 (risky)==</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>==Solution 3 (risky)==</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>
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<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;">==Video Solution==</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;">https://youtu.be/diLCwN-358s</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>== 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>Daijobuhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=122494&oldid=prevJbala at 23:33, 15 May 20202020-05-15T23:33:57Z<p></p>
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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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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;">[[Category:Intermediate Geometry Problems]]</ins></div></td></tr>
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</table>Jbalahttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=110203&oldid=prevToinfinity: /* Solution 1 */2019-10-09T03:00:25Z<p><span dir="auto"><span class="autocomment">Solution 1</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 03:00, 9 October 2019</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><cmath>  (0,0), \left(1, \frac{3}{7}\right), \left(2, \frac{6}{7}\right), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), \left(5, 2 + \frac{1}{7}\right), \left(6, 2 + \frac{4}{7}\right), (7,3). </cmath></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><cmath>  (0,0), \left(1, \frac{3}{7}\right), \left(2, \frac{6}{7}\right), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), \left(5, 2 + \frac{1}{7}\right), \left(6, 2 + \frac{4}{7}\right), (7,3). </cmath></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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>\left(2 + \frac{1}{10},  1 - \frac{1}{10}\right) = \left(\frac{21}{10}, \frac{9}{10}\right)</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>\left(2 + \frac{1}{10},  1 - \frac{1}{10}\right) = \left(\frac{21}{10}, \frac{9}{10}\right)</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math><ins class="diffchange diffchange-inline">\left</ins>(\frac{21}{10}, \frac{9}{10}<ins class="diffchange diffchange-inline">\right</ins>)</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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;"><div>(Solution by gundraja)</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>(Solution by gundraja)</div></td></tr>
</table>Toinfinityhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=110202&oldid=prevToinfinity: /* Solution 1 */2019-10-09T02:59:46Z<p><span dir="auto"><span class="autocomment">Solution 1</span></span></p>
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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><cmath>  (0,0), \left(1, \frac{3}{7}\right), \left(2, \frac{6}{7}\right), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), \left(5, 2 + \frac{1}{7}\right), \left(6, 2 + \frac{4}{7}\right), (7,3). </cmath></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><cmath>  (0,0), \left(1, \frac{3}{7}\right), \left(2, \frac{6}{7}\right), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), \left(5, 2 + \frac{1}{7}\right), \left(6, 2 + \frac{4}{7}\right), (7,3). </cmath></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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>(2 + \frac{1}{10},  1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math><ins class="diffchange diffchange-inline">\left</ins>(2 + \frac{1}{10},  1 - \frac{1}{10}<ins class="diffchange diffchange-inline">\right</ins>) = <ins class="diffchange diffchange-inline">\left</ins>(\frac{21}{10}, \frac{9}{10}<ins class="diffchange diffchange-inline">\right</ins>)</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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;"><div>(Solution by gundraja)</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>(Solution by gundraja)</div></td></tr>
</table>Toinfinityhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=110201&oldid=prevToinfinity: /* Solution 1 */2019-10-09T02:59:10Z<p><span dir="auto"><span class="autocomment">Solution 1</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 02:59, 9 October 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</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>First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</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>First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</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><cmath>  (0,0), (1, \frac{3}{7}), (2, \frac{6}{7}), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), (5, 2 + \frac{1}{7}), (6, 2 + \frac{4}{7}), (7,3). </cmath></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><cmath>  (0,0), <ins class="diffchange diffchange-inline">\left</ins>(1, \frac{3}{7}<ins class="diffchange diffchange-inline">\right</ins>), <ins class="diffchange diffchange-inline">\left</ins>(2, \frac{6}{7}<ins class="diffchange diffchange-inline">\right</ins>), \left(3, 1 + \frac{2}{7}\right), \left(4, 1 + \frac{5}{7}\right), <ins class="diffchange diffchange-inline">\left</ins>(5, 2 + \frac{1}{7}<ins class="diffchange diffchange-inline">\right</ins>), <ins class="diffchange diffchange-inline">\left</ins>(6, 2 + \frac{4}{7}<ins class="diffchange diffchange-inline">\right</ins>), (7,3). </cmath></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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>(2 + \frac{1}{10},  1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>(2 + \frac{1}{10},  1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</math>.</div></td></tr>
</table>Toinfinityhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=110200&oldid=prevToinfinity: /* Solution 1 */2019-10-09T02:58:32Z<p><span dir="auto"><span class="autocomment">Solution 1</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 02:58, 9 October 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</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>First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</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>First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</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><cmath>  (0,0), (1, \frac{3}{7}), (2, \frac{6}{7}), (3, 1 + \frac{2}{7}), (4, 1 + \frac{5}{7}), (5, 2 + \frac{1}{7}), (6, 2 + \frac{4}{7}), (7,3). </cmath></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><cmath>  (0,0), (1, \frac{3}{7}), (2, \frac{6}{7}), <ins class="diffchange diffchange-inline">\left</ins>(3, 1 + \frac{2}{7}<ins class="diffchange diffchange-inline">\right</ins>), <ins class="diffchange diffchange-inline">\left</ins>(4, 1 + \frac{5}{7}<ins class="diffchange diffchange-inline">\right</ins>), (5, 2 + \frac{1}{7}), (6, 2 + \frac{4}{7}), (7,3). </cmath></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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>(2 + \frac{1}{10},  1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</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>We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>.  Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates <math>(2 + \frac{1}{10},  1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})</math>.  Because <math>\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}</math>, <math>(\frac{21}{10}, \frac{9}{10})</math> lies on <math>l</math>.  Since the circle centered at <math>(2,1)</math> is contained inside the square, this circle does not intersect <math>l</math>.  Similarly the upper left vertex of the square centered at <math>(5,2)</math> is on <math>l</math>.  Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between <math>(0,0)</math> and <math>(7,3)</math> that intersect <math>l</math>.  Since there are <math>\frac{1001}{7} = \frac{429}{3} = 143</math> segments from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>, the above count is yields <math>143 \cdot 2 = 286</math> squares.  Since every lattice point on <math>l</math> is of the form <math>(3k, 7k)</math> where <math>0 \le k \le 143</math>, there are <math>144</math> lattice points on <math>l</math>. Centered at each lattice point, there is one square and one circle, hence this counts <math>288</math> squares and circles.  Thus <math>m + n = 286 + 288 = \boxed{574}</math>.</div></td></tr>
</table>Toinfinityhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=103657&oldid=prevFirebolt360: /* See also */2019-02-21T00:21:22Z<p><span dir="auto"><span class="autocomment">See also</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 00:21, 21 February 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32" >Line 32:</td>
<td colspan="2" class="diff-lineno">Line 32:</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>==Solution 3 (risky)==</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>==Solution 3 (risky)==</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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 style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">== See also ==</del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</del></div></td><td colspan="2"> </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 style="font-weight: bold; text-decoration: none;">{{MAA Notice}}</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;"></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>== 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="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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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>Firebolt360https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=103656&oldid=prevFirebolt360: /* Solution 3 (risky) */2019-02-21T00:20:53Z<p><span dir="auto"><span class="autocomment">Solution 3 (risky)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 00:20, 21 February 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32" >Line 32:</td>
<td colspan="2" class="diff-lineno">Line 32:</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>==Solution 3 (risky)==</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>==Solution 3 (risky)==</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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;">== See also ==</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;">{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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;">{{MAA Notice}}</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>== 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="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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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>{{AIME box|year=2016|n=I|num-b=13|num-a=15}}</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>Firebolt360https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_14&diff=103655&oldid=prevFirebolt360: /* Solution 3 (risky) */2019-02-21T00:20:43Z<p><span dir="auto"><span class="autocomment">Solution 3 (risky)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<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 00:20, 21 February 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l30" >Line 30:</td>
<td colspan="2" class="diff-lineno">Line 30:</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>-Patrick4President</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>-Patrick4President</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>=Solution 3 (risky)=</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">=</ins>=Solution 3 (risky)<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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>When drawing a diagram, one might notice that the line might intersect the corners of the squares. Keeping in mind that the problem is number 14, one can assume that the line does indeed intersect the corners of the square, otherwise, the problem would be too easy. Thus, we can say that in a box from <math>(0,0)</math> to <math>(7,3)</math>, there are 4 squares intercepted, and 2 circles intercepted. Ignoring the square and circle on <math>(0,0)</math>, we can repeat the segment <math>143</math> times. <math>143\cdot4 = 572</math> adding back in the square and the circle, we get <math>\boxed{574}</math> as the answer. This is risky, so do not use this unless you have no other possible strategies to tackle this problem.</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>Firebolt360