https://artofproblemsolving.com/wiki/index.php?title=1969_IMO_Problems/Problem_1&feed=atom&action=history
1969 IMO Problems/Problem 1 - Revision history
2024-03-28T22:07:09Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=1969_IMO_Problems/Problem_1&diff=183732&oldid=prev
Mathboy100 at 22:13, 7 December 2022
2022-12-07T22:13:29Z
<p></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 22:13, 7 December 2022</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;"></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>==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>==Solution==</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">The equation was </del><math><del class="diffchange diffchange-inline">z </del>= <del class="diffchange diffchange-inline">n</del>^4 <del class="diffchange diffchange-inline">+ </del>a</math> <del class="diffchange diffchange-inline">,you can put </del><math>a <del class="diffchange diffchange-inline">= 4 m^4 </del></math> <del class="diffchange diffchange-inline">for all natural numbers m</del>. <del class="diffchange diffchange-inline">So you will get </del><math> <del class="diffchange diffchange-inline">z = </del>n^4 + <del class="diffchange diffchange-inline">4 m</del>^4 <del class="diffchange diffchange-inline">= </del>n^4+<del class="diffchange diffchange-inline">4m</del>^4 +4n^2 <del class="diffchange diffchange-inline">m</del>^<del class="diffchange diffchange-inline">2 </del>- 4n^<del class="diffchange diffchange-inline">2 m</del>^2</<del class="diffchange diffchange-inline">math</del>> <<del class="diffchange diffchange-inline">math</del>><del class="diffchange diffchange-inline">z </del>= (n^2+<del class="diffchange diffchange-inline">2 m</del>^2)^2 - (<del class="diffchange diffchange-inline">2nm</del>)^2 = (n^2+<del class="diffchange diffchange-inline">2 m</del>^2 -<del class="diffchange diffchange-inline">2nm</del>)(n^2+2 <del class="diffchange diffchange-inline">m</del>^2 +<del class="diffchange diffchange-inline">2nm) </del></math> <del class="diffchange diffchange-inline">so you get </del><math><del class="diffchange diffchange-inline">z</del></math> <del class="diffchange diffchange-inline">is composite </del>for all <math> <del class="diffchange diffchange-inline">a = 4 m</del>^4</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">Suppose that </ins><math><ins class="diffchange diffchange-inline">a </ins>= <ins class="diffchange diffchange-inline">4k</ins>^4<ins class="diffchange diffchange-inline"></math> for some <math></ins>a</math><ins class="diffchange diffchange-inline">. We will prove that </ins><math>a</math> <ins class="diffchange diffchange-inline">satisfies the property outlined above</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> </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 class="diffchange diffchange-inline">The polynomial </ins><math>n^4 + <ins class="diffchange diffchange-inline">4k</ins>^4<ins class="diffchange diffchange-inline"></math> can be factored as follows:</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> </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 class="diffchange diffchange-inline"><cmath></ins>n^4 + <ins class="diffchange diffchange-inline">4k^4</cmath></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 class="diffchange diffchange-inline"><cmath> = n</ins>^4 + 4n<ins class="diffchange diffchange-inline">^2k</ins>^2 <ins class="diffchange diffchange-inline">+ 4k</ins>^<ins class="diffchange diffchange-inline">4 </ins>- 4n^<ins class="diffchange diffchange-inline">2k</ins>^2</<ins class="diffchange diffchange-inline">cmath</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 class="diffchange diffchange-inline">cmath</ins>> = (n^2 + <ins class="diffchange diffchange-inline">2k</ins>^2)^2 - (<ins class="diffchange diffchange-inline">2nk</ins>)^2<ins class="diffchange diffchange-inline"></cmath></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 class="diffchange diffchange-inline"><cmath> </ins>= (n^2 + <ins class="diffchange diffchange-inline">2k</ins>^2 - <ins class="diffchange diffchange-inline">2nk</ins>)(n^2 + <ins class="diffchange diffchange-inline">2k^</ins>2 <ins class="diffchange diffchange-inline">+ 2nk)</cmath></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> </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 class="diffchange diffchange-inline">Both factors are positive, because if the left one is negative, then the right one would also negative, which is clearly false.</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> </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 class="diffchange diffchange-inline">It is also simple to prove that <math>n</ins>^2 + <ins class="diffchange diffchange-inline">2k^2 - 2nk > 1</ins></math> <ins class="diffchange diffchange-inline">when </ins><math><ins class="diffchange diffchange-inline">k > 1</ins></math><ins class="diffchange diffchange-inline">. Thus, </ins>for all <math><ins class="diffchange diffchange-inline">k > 2</math>, <math>4k</ins>^4</math> <ins class="diffchange diffchange-inline">is a valid value of <math>a</math>, completing the proof. <math>\square</math></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> </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 class="diffchange diffchange-inline">~mathboy100</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>{{alternate 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>{{alternate 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;"></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 == {{IMO box|year=1969|before=First question|num-a=2}}</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 == {{IMO box|year=1969|before=First question|num-a=2}}</div></td></tr>
</table>
Mathboy100
https://artofproblemsolving.com/wiki/index.php?title=1969_IMO_Problems/Problem_1&diff=143786&oldid=prev
Hamstpan38825 at 17:35, 29 January 2021
2021-01-29T17:35:46Z
<p></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 17:35, 29 January 2021</td>
</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 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>The equation was<math>z = n^4 + a</math> ,you can put <math> a = 4 m^4 </math> for all natural numbers m. So you will get <math> z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2</math> <math>z = (n^2+2 m^2)^2 - (2nm)^2 = (n^2+2 m^2 -2nm)(n^2+2 m^2 +2nm) </math> so you get z is composite for all <math> a = 4 m^4</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">==Problem==</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 class="diffchange diffchange-inline">Prove that there are infinitely many natural numbers <math>a</math> with the following property: the number <math>z = n^4 + a</math> is not prime for any natural number <math>n</math>.</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> </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 class="diffchange diffchange-inline">==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>The equation was <math>z = n^4 + a</math> ,you can put <math>a = 4 m^4 </math> for all natural numbers m. So you will get <math> z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2</math> <math>z = (n^2+2 m^2)^2 - (2nm)^2 = (n^2+2 m^2 -2nm)(n^2+2 m^2 +2nm) </math> so you get <ins class="diffchange diffchange-inline"><math></ins>z<ins class="diffchange diffchange-inline"></math> </ins>is composite for all <math> a = 4 m^4</math></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> </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 class="diffchange diffchange-inline">{{alternate solutions}}</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> </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 class="diffchange diffchange-inline">== See Also == {{IMO box|year=1969|before=First question|num-a=2}}</ins></div></td></tr>
</table>
Hamstpan38825
https://artofproblemsolving.com/wiki/index.php?title=1969_IMO_Problems/Problem_1&diff=36606&oldid=prev
Thrax: Created page with 'The equation was<math>z = n^4 + a</math> ,you can put <math> a = 4 m^4 </math> for all natural numbers m. So you will get <math> z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2</…'
2011-02-07T22:26:43Z
<p>Created page with 'The equation was<math>z = n^4 + a</math> ,you can put <math> a = 4 m^4 </math> for all natural numbers m. So you will get <math> z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2</…'</p>
<p><b>New page</b></p><div>The equation was<math>z = n^4 + a</math> ,you can put <math> a = 4 m^4 </math> for all natural numbers m. So you will get <math> z = n^4 + 4 m^4 = n^4+4m^4 +4n^2 m^2 - 4n^2 m^2</math> <math>z = (n^2+2 m^2)^2 - (2nm)^2 = (n^2+2 m^2 -2nm)(n^2+2 m^2 +2nm) </math> so you get z is composite for all <math> a = 4 m^4</math></div>
Thrax