https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&feed=atom&action=history Chittur Gopalakrishnavishwanathasrinivasaiyer Lemma - Revision history 2022-07-02T10:20:00Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&diff=162526&oldid=prev Mathsweat notreally at 00:59, 21 September 2021 2021-09-21T00:59:10Z <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 00:59, 21 September 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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;42\, \cdot&lt;/math&gt; &lt;math&gt;\text{mod} \sqrt{4761}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(42*10\cdot(70-1)^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{23\cdot3}.&lt;/math&gt; My <del class="diffchange diffchange-inline">coach </del>Iyer Sir approved this nice lemma.</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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;42\, \cdot&lt;/math&gt; &lt;math&gt;\text{mod} \sqrt{4761}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(42*10\cdot(70-1)^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{23\cdot3}.&lt;/math&gt; My <ins class="diffchange diffchange-inline">coaches </ins>Iyer Sir <ins class="diffchange diffchange-inline">and Barnes </ins>approved this nice lemma.</div></td></tr> </table> Mathsweat notreally https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&diff=153019&oldid=prev Readyplayerone at 01:29, 1 May 2021 2021-05-01T01:29:55Z <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 01:29, 1 May 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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;\text{<del class="diffchange diffchange-inline">42*10 </del>mod <del class="diffchange diffchange-inline">(</del>\sqrt{4761}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(42*10\cdot(70-1)^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{23\cdot3}.&lt;/math&gt; My coach Iyer Sir approved this nice lemma.</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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to <ins class="diffchange diffchange-inline">&lt;math&gt;42\, \cdot&lt;/math&gt; </ins>&lt;math&gt;\text{mod<ins class="diffchange diffchange-inline">} </ins>\sqrt{4761}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(42*10\cdot(70-1)^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{23\cdot3}.&lt;/math&gt; My coach Iyer Sir approved this nice lemma.</div></td></tr> </table> Readyplayerone https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&diff=153018&oldid=prev Readyplayerone at 01:28, 1 May 2021 2021-05-01T01:28:15Z <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 01:28, 1 May 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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;\text{<del class="diffchange diffchange-inline">420 </del>mod <del class="diffchange diffchange-inline">69</del>}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(<del class="diffchange diffchange-inline">420</del>\<del class="diffchange diffchange-inline">cdot69</del>^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{<del class="diffchange diffchange-inline">69</del>}.&lt;/math&gt; My coach Iyer Sir approved this nice lemma.</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>&#160; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;\text{<ins class="diffchange diffchange-inline">42*10 </ins>mod <ins class="diffchange diffchange-inline">(\sqrt{4761</ins>}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(<ins class="diffchange diffchange-inline">42*10</ins>\<ins class="diffchange diffchange-inline">cdot(70-1)</ins>^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{<ins class="diffchange diffchange-inline">23\cdot3</ins>}.&lt;/math&gt; My coach Iyer Sir approved this nice lemma.</div></td></tr> </table> Readyplayerone https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&diff=153017&oldid=prev Readyplayerone: Created page with " We know that an $x$ exists that equal to $\text{420 mod 69}.$ This $x$ is very powerful in competition math problems. Usually coming up on JM..." 2021-05-01T01:26:56Z <p>Created page with &quot; We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;\text{420 mod 69}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JM...&quot;</p> <p><b>New page</b></p><div> We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;\text{420 mod 69}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(420\cdot69^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{69}.&lt;/math&gt; My coach Iyer Sir approved this nice lemma.</div> Readyplayerone