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	<title>Project PEN</title>
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	<description>Problems in Elementary Number Theory</description>
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		<title>PEN Global</title>
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		<pubDate>Tue, 23 Jun 2009 13:17:56 +0000</pubDate>
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		<description><![CDATA[Anyone who wants to contribute the PEN Global, the translation work of Problems in Elementary Number Theory 2 [ 2009 ] No. 1, please submit your translation to Yimin Ge at together with your LaTeX file, its compiled PDF file and fonts file. You have to use the TeX file available here. Download gPENv02n1eng.tex. The [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=433&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>Anyone who wants to contribute the PEN Global, the translation work of <strong><a href="http://projectpen.wordpress.com/all-issues/">Problems in Elementary Number Theory 2 [ 2009 ] No. 1</a></strong>, please submit your translation to <strong>Yimin Ge</strong> at<strong> <span><img src='http://s0.wp.com/latex.php?latex=%5Ctext%7B%5Csmall+yimin.ge%40gmx.at%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='&#92;text{&#92;small yimin.ge@gmx.at}' title='&#92;text{&#92;small yimin.ge@gmx.at}' class='latex' /> </span></strong> together with your <strong>LaTeX file,</strong> its <strong>compiled PDF file </strong>and <strong>fonts file</strong>.</p>
<p>You have to use the TeX file available <strong><a href="http://projectpen.wordpress.com/global/">here</a></strong>. Download gPENv02n1eng.tex. <strong>The deadline for your submission is 15th, July.</strong>.</p>
<p>Before you contact <strong>Yimin Ge</strong>, please have a look at the PEN Global page at <a href="http://projectpen.wordpress.com/global">http://projectpen.wordpress.com/global</a> .</p>
<p><strong>Notice:</strong> We&#8217;ve already find translators: in Vietnamese,  Greek, Spanish, Bangla, Bosnian, Chinese, Croatian, Serbian [Latin] and Serbian [Cyrillic]</p>
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		<title>Editors-in-Chief for 2010</title>
		<link>http://projectpen.wordpress.com/2009/06/22/editors-in-chief-for-2010/</link>
		<comments>http://projectpen.wordpress.com/2009/06/22/editors-in-chief-for-2010/#comments</comments>
		<pubDate>Mon, 22 Jun 2009 14:21:07 +0000</pubDate>
		<dc:creator>compactorange</dc:creator>
				<category><![CDATA[Notice]]></category>

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		<description><![CDATA[As you know, Problems in Elementary Number Theory 2 [ 2009 ] No. 1 is NOW online. The Editors-in-Chief of the next issue are Daniel Kohen (Argentina) Cosmin Pohoata (Romania) Harun Šiljak (Bosnia and Herzegovina) Peter Vandendriessche (Belgium) Anyone who wants to be one of the Editors-in-Chief of the fourth and fifth issue of Problems [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=424&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>As you know, <span style="color:#008000;"><a href="http://projectpen.wordpress.com/all-issues/"><strong>Problems in Elementary Number Theory 2 [ 2009 ] No. 1</strong></a></span> is NOW online. The <strong>Editors-in-Chief </strong>of the next issue are</p>
<p><span style="color:#008000;"><strong>Daniel Kohen</strong> </span>(Argentina)<br />
<span style="color:#008000;"><strong>Cosmin Pohoata</strong></span> (Romania)<br />
<span style="color:#008000;"><strong>Harun Šiljak </strong> </span> (Bosnia and Herzegovina)<br />
<span style="color:#008000;"><strong>Peter Vandendriessche</strong> </span>(Belgium)</p>
<p>Anyone who wants to be one of the <strong>Editors-in-Chief </strong>of the fourth and fifth issue of <strong>Problems in Elementary Number Theory</strong> (2010), <span style="color:#0000ff;"><strong>APPLY NOW</strong>.</span></p>
<p>Please, send an email to us at <strong>pen@problem-solving.be</strong> with your brief C.V.</p>
<p>You need to know basic <strong>LaTeX skills</strong> and of course you have to be<strong> fluent with Olympiad-style problems</strong> from Elementary number theory.</p>
<p>We are planning to recruit two or three <strong>Editors-in-Chief</strong> for <strong>Problems in Elementary Number Theory</strong> 2010. <strong><span style="color:#0000ff;">The deadline for application is 31th, July. </span></strong></p>
<p>from Hojoo Lee (the founder of project PEN)</p>
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		<title>Problems in Elementary Number Theory 2 [ 2009 ] No. 1</title>
		<link>http://projectpen.wordpress.com/2009/06/19/problems-in-elementary-number-theory-1-2009-no-2/</link>
		<comments>http://projectpen.wordpress.com/2009/06/19/problems-in-elementary-number-theory-1-2009-no-2/#comments</comments>
		<pubDate>Fri, 19 Jun 2009 08:26:01 +0000</pubDate>
		<dc:creator>compactorange</dc:creator>
				<category><![CDATA[Notice]]></category>

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		<description><![CDATA[Is now available here ! This weblication is written by PEN team and contributors: Alexander Remorov (Canada), Darij Grinberg (Germany), Harun Siljak (Bosnia and Herzegovina), Marin Misur (Croatia). We also need to thank  two Editors-in-Chief: Daniel Kohen (Argentina) and Cosmin Pohoata (Romania). Here goes the table of contents: 1 Problems 1 2 Articles 3 2.1 Three Ways to [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=418&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>Is <strong>now</strong> available <a href="http://projectpen.wordpress.com/all-issues"><strong>here</strong></a> !</p>
<p>This weblication is written by PEN team and contributors:</p>
<p><strong>Alexander Remorov</strong> (Canada),<br />
<strong>Darij Grinberg</strong> (Germany),<br />
<strong>Harun Siljak</strong> (Bosnia and Herzegovina),<br />
<strong>Mari</strong><strong>n Misur</strong> (Croatia).</p>
<p>We also need to thank  two Editors-in-Chief:</p>
<p><strong>Daniel Kohen</strong> (Argentina) and <strong>Cosmin Pohoata</strong> (Romania).</p>
<p><strong><span style="color:#0000ff;">Here goes the table of contents: </span></strong></p>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">1 Problems 1</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2 Articles 3</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.4 Dierent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.8 Fractions Mod p and Wolstenholme&#8217;s theorem . . . . . . . . . . . . . . . . . . . . . 29</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33</div>
<div id="_mcePaste" style="position:absolute;left:-10000px;top:0;width:1px;height:1px;">2.10 Sequences of Consecutive Integers . . . . . .</div>
<div>1 Problems 1</div>
<div>2 Articles 3</div>
<div>2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3</div>
<div>2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8</div>
<div>2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11</div>
<div>2.4 Dierent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15</div>
<div>2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19</div>
<div>2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24</div>
<div>2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27</div>
<div>2.8 Fractions Mod p and Wolstenholme&#8217;s theorem . . . . . . . . . . . . . . . . . . . . . 29</div>
<div>2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33</div>
<div>2.10 Sequences of Consecutive Integers . . . . . .</div>
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		<title>20S. Sequences of Consecutive Integers</title>
		<link>http://projectpen.wordpress.com/2009/06/16/20s-sequences-of-consecutive-integers/</link>
		<comments>http://projectpen.wordpress.com/2009/06/16/20s-sequences-of-consecutive-integers/#comments</comments>
		<pubDate>Tue, 16 Jun 2009 16:08:13 +0000</pubDate>
		<dc:creator>danielkohen</dc:creator>
				<category><![CDATA[PEN Solutions Archive]]></category>

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		<description><![CDATA[A37 If is a natural number, prove that the number is not a perfect square. A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others. O51 Prove the among consecutive integers it is always possible to find one which is relatively prime to all [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=407&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?t=150405">A37</a></strong> <span>If <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='n' title='n' class='latex' /> is a natural number, prove that the number</span></p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=%28n+%2B+1%29%28n+%2B+2%29%5Ccdots%28n+%2B+10%29&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='(n + 1)(n + 2)&#92;cdots(n + 10)' title='(n + 1)(n + 2)&#92;cdots(n + 10)' class='latex' /></p>
<p>is not a perfect square.</p>
<div>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?t=150377">A9</a></strong> <span>Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.<span><br />
</span></span></p>
</div>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=850004#850004">O51</a></strong> <span>Prove the among <img src='http://s0.wp.com/latex.php?latex=16&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='16' title='16' class='latex' /> consecutive integers it is always possible to find one which is relatively prime to all the rest.</span></p>
<p style="text-align:left;"><span>Here is the official solution file: <a href="http://projectpen.files.wordpress.com/2009/06/pen-a9-a37-o511.pdf">PEN A9 A37 O51</a></span></p>
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			<media:title type="html">danielkohen</media:title>
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		<title>20. Sequences of Consecutive Integers</title>
		<link>http://projectpen.wordpress.com/2009/05/04/398/</link>
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		<pubDate>Mon, 04 May 2009 15:17:05 +0000</pubDate>
		<dc:creator>cpohoata</dc:creator>
				<category><![CDATA[Problem of the Bi-Week]]></category>

		<guid isPermaLink="false">http://projectpen.wordpress.com/?p=398</guid>
		<description><![CDATA[A37 If is a natural number, prove that the number is not a perfect square. A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others. O51 Prove the among consecutive integers it is always possible to find one which is relatively prime to all [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=398&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?t=150405">A37</a></strong> <span class="postbody">If <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='n' title='n' class='latex' /> is a natural number, prove that the number</span></p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=%28n+%2B+1%29%28n+%2B+2%29%5Ccdots%28n+%2B+10%29&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='(n + 1)(n + 2)&#92;cdots(n + 10)' title='(n + 1)(n + 2)&#92;cdots(n + 10)' class='latex' /></p>
<p>is not a perfect square.</p>
<div>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?t=150377">A9</a></strong> <span class="postdetails">Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.<span class="postdetails"><br />
</span></span></p>
</div>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=850004#850004">O51</a></strong> <span class="postdetails">Prove the among <img src='http://s0.wp.com/latex.php?latex=16&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='16' title='16' class='latex' /> consecutive integers it is always possible to find one which is relatively prime to all the rest.<span class="postdetails"><br />
</span> </span></p>
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			<media:title type="html">cpohoata</media:title>
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		<title>19S. Binomial Sum Divisible by Primes</title>
		<link>http://projectpen.wordpress.com/2009/04/25/19s-binomial-sum-divisible-by-primes/</link>
		<comments>http://projectpen.wordpress.com/2009/04/25/19s-binomial-sum-divisible-by-primes/#comments</comments>
		<pubDate>Sat, 25 Apr 2009 22:03:21 +0000</pubDate>
		<dc:creator>cpohoata</dc:creator>
				<category><![CDATA[PEN Solutions Archive]]></category>

		<guid isPermaLink="false">http://projectpen.wordpress.com/?p=388</guid>
		<description><![CDATA[E16 Prove that for any prime in the interval , divides Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=388&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849498#849498">E16</a></strong> Prove that for any prime <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p' title='p' class='latex' /> in the interval <img src='http://s0.wp.com/latex.php?latex=%5Cleft%5Dn%2C+%5Cfrac+%7B4n%7D%7B3%7D%5Cright%5D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='&#92;left]n, &#92;frac {4n}{3}&#92;right]' title='&#92;left]n, &#92;frac {4n}{3}&#92;right]' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p' title='p' class='latex' /> divides</p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=%5Csum%5E%7Bn%7D_%7Bj+%3D+0%7D%7B%7Bn%7D%5Cchoose%7Bj%7D%7D%5E%7B4%7D.&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='&#92;sum^{n}_{j = 0}{{n}&#92;choose{j}}^{4}.' title='&#92;sum^{n}_{j = 0}{{n}&#92;choose{j}}^{4}.' class='latex' /></p>
<p>Here is the official solution file (written by <strong>Darij Grinberg</strong>, Germany): <a href='http://projectpen.files.wordpress.com/2009/04/pen-e16.pdf'>pen19.pdf</a>.</p>
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			<media:title type="html">cpohoata</media:title>
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		<title>19. Binomial Sum Divisible by Primes</title>
		<link>http://projectpen.wordpress.com/2009/04/20/365/</link>
		<comments>http://projectpen.wordpress.com/2009/04/20/365/#comments</comments>
		<pubDate>Mon, 20 Apr 2009 10:26:08 +0000</pubDate>
		<dc:creator>cpohoata</dc:creator>
				<category><![CDATA[Problem of the Bi-Week]]></category>

		<guid isPermaLink="false">http://projectpen.wordpress.com/?p=365</guid>
		<description><![CDATA[E16 Prove that for any prime in the interval , divides<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=365&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849498#849498">E16</a></strong> Prove that for any prime <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p' title='p' class='latex' /> in the interval <img src='http://s0.wp.com/latex.php?latex=%5Cleft%5Dn%2C+%5Cfrac+%7B4n%7D%7B3%7D%5Cright%5D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='&#92;left]n, &#92;frac {4n}{3}&#92;right]' title='&#92;left]n, &#92;frac {4n}{3}&#92;right]' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p' title='p' class='latex' /> divides</p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=%5Csum%5E%7Bn%7D_%7Bj+%3D+0%7D%7B%7Bn%7D%5Cchoose%7Bj%7D%7D%5E%7B4%7D.&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='&#92;sum^{n}_{j = 0}{{n}&#92;choose{j}}^{4}.' title='&#92;sum^{n}_{j = 0}{{n}&#92;choose{j}}^{4}.' class='latex' /></p>
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			<media:title type="html">cpohoata</media:title>
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		<title>18S.Fractions Mod p and Wolstenholme&#8217;s Theorem</title>
		<link>http://projectpen.wordpress.com/2009/04/08/18sfractions-mod-p-and-wolstenholmes-theorem/</link>
		<comments>http://projectpen.wordpress.com/2009/04/08/18sfractions-mod-p-and-wolstenholmes-theorem/#comments</comments>
		<pubDate>Wed, 08 Apr 2009 01:30:18 +0000</pubDate>
		<dc:creator>danielkohen</dc:creator>
				<category><![CDATA[PEN Solutions Archive]]></category>

		<guid isPermaLink="false">http://projectpen.wordpress.com/?p=360</guid>
		<description><![CDATA[A23 Prove that if is expressed as a fraction, where is a prime, then divides the numerator. A24 Let be a prime number and . Prove that is divisible by . Here is the official solution file: pen18.pdf<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=360&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<div>
<div>
<div>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849236#849236">A23</a></strong> <span class="postbody">Prove that if </span></p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7D%2B%5Ccdots%2B%5Cfrac%7B1%7D%7Bp-1%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='1+&#92;frac{1}{2}+&#92;frac{1}{3}+&#92;cdots+&#92;frac{1}{p-1}' title='1+&#92;frac{1}{2}+&#92;frac{1}{3}+&#92;cdots+&#92;frac{1}{p-1}' class='latex' /></p>
<div>
<div>
<p>is expressed as a fraction, where <img src='http://s0.wp.com/latex.php?latex=p%3E3&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p&gt;3' title='p&gt;3' class='latex' /> is a prime, then <img src='http://s0.wp.com/latex.php?latex=p%5E2&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p^2' title='p^2' class='latex' /> divides the numerator.</p>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849240#849240">A24</a></strong> <span class="postdetails">Let <img src='http://s0.wp.com/latex.php?latex=p%3E3&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p&gt;3' title='p&gt;3' class='latex' /> be a prime number and <img src='http://s0.wp.com/latex.php?latex=k%3D%5Clfloor%5Cfrac%7B2p%7D%7B3%7D%5Crfloor&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='k=&#92;lfloor&#92;frac{2p}{3}&#92;rfloor' title='k=&#92;lfloor&#92;frac{2p}{3}&#92;rfloor' class='latex' />. Prove that</span></p>
<p style="text-align:center;"><span class="postdetails"><img src='http://s0.wp.com/latex.php?latex=%7Bp+%5Cchoose+1%7D%2B%7Bp+%5Cchoose+2%7D%2B%5Ccdots%2B%7Bp+%5Cchoose+k%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='{p &#92;choose 1}+{p &#92;choose 2}+&#92;cdots+{p &#92;choose k}' title='{p &#92;choose 1}+{p &#92;choose 2}+&#92;cdots+{p &#92;choose k}' class='latex' /></span></p>
<p style="text-align:left;"><span class="postdetails">is divisible by <img src='http://s0.wp.com/latex.php?latex=p%5E2&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p^2' title='p^2' class='latex' />.</span></p>
</div>
</div>
</div>
</div>
</div>
<p>Here is the official solution file: <a href='http://projectpen.files.wordpress.com/2009/04/pen-a23-a24-version-edited2.pdf'>pen18.pdf</a></p>
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			<media:title type="html">danielkohen</media:title>
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		<title>18. Fractions Mod p and Wolstenholme&#8217;s Theorem</title>
		<link>http://projectpen.wordpress.com/2009/04/03/18-fractions-mod-p-and-wolstenholmes-theorem/</link>
		<comments>http://projectpen.wordpress.com/2009/04/03/18-fractions-mod-p-and-wolstenholmes-theorem/#comments</comments>
		<pubDate>Fri, 03 Apr 2009 20:00:49 +0000</pubDate>
		<dc:creator>danielkohen</dc:creator>
				<category><![CDATA[Problem of the Bi-Week]]></category>

		<guid isPermaLink="false">http://projectpen.wordpress.com/?p=352</guid>
		<description><![CDATA[A23 Prove that if is expressed as a fraction, where is a prime, then divides the numerator. A24 Let be a prime number and . Prove that is divisible by .<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=352&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849236#849236">A23</a></strong> <span class="postbody">Prove that if </span></p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7D%2B%5Ccdots%2B%5Cfrac%7B1%7D%7Bp-1%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='1+&#92;frac{1}{2}+&#92;frac{1}{3}+&#92;cdots+&#92;frac{1}{p-1}' title='1+&#92;frac{1}{2}+&#92;frac{1}{3}+&#92;cdots+&#92;frac{1}{p-1}' class='latex' /></p>
<div>
<div>
<p>is expressed as a fraction, where <img src='http://s0.wp.com/latex.php?latex=p%3E3&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p&gt;3' title='p&gt;3' class='latex' /> is a prime, then <img src='http://s0.wp.com/latex.php?latex=p%5E2&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p^2' title='p^2' class='latex' /> divides the numerator.</p>
<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?p=849240#849240">A24</a></strong> <span class="postdetails">Let <img src='http://s0.wp.com/latex.php?latex=p%3E3&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p&gt;3' title='p&gt;3' class='latex' /> be a prime number and <img src='http://s0.wp.com/latex.php?latex=k%3D%5Clfloor%5Cfrac%7B2p%7D%7B3%7D%5Crfloor&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='k=&#92;lfloor&#92;frac{2p}{3}&#92;rfloor' title='k=&#92;lfloor&#92;frac{2p}{3}&#92;rfloor' class='latex' />. Prove that</span></p>
<p style="text-align:center;"><span class="postdetails"><img src='http://s0.wp.com/latex.php?latex=%7Bp+%5Cchoose+1%7D%2B%7Bp+%5Cchoose+2%7D%2B%5Ccdots%2B%7Bp+%5Cchoose+k%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='{p &#92;choose 1}+{p &#92;choose 2}+&#92;cdots+{p &#92;choose k}' title='{p &#92;choose 1}+{p &#92;choose 2}+&#92;cdots+{p &#92;choose k}' class='latex' /></span><span class="postdetails"></span></p>
<p style="text-align:left;"><span class="postdetails">is divisible by <img src='http://s0.wp.com/latex.php?latex=p%5E2&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p^2' title='p^2' class='latex' />.</span></p>
<p><span class="postdetails"><br />
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		<title>17S. A Hidden Divisibility</title>
		<link>http://projectpen.wordpress.com/2009/03/16/334/</link>
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		<pubDate>Mon, 16 Mar 2009 12:15:04 +0000</pubDate>
		<dc:creator>cpohoata</dc:creator>
				<category><![CDATA[PEN Solutions Archive]]></category>

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		<description><![CDATA[A13 Show that for all prime numbers , is an integer. Here is the official solution file: pen17.pdf.<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=projectpen.wordpress.com&amp;blog=1775818&amp;post=334&amp;subd=projectpen&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p style="text-align:left;"><strong><a href="http://www.mathlinks.ro/viewtopic.php?t=150381">A13</a></strong> Show that for all prime numbers <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='p' title='p' class='latex' />,</p>
<p style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex=Q%28p%29+%3D+%5Cprod%5E%7Bp+-+1%7D_%7Bk+%3D+1%7Dk%5E%7B2k+-+p+-+1%7D&amp;bg=ffffff&amp;fg=333333&amp;s=0' alt='Q(p) = &#92;prod^{p - 1}_{k = 1}k^{2k - p - 1}' title='Q(p) = &#92;prod^{p - 1}_{k = 1}k^{2k - p - 1}' class='latex' /></p>
<p>is an integer.</p>
<p>Here is the official solution file: <a href="http://projectpen.files.wordpress.com/2009/03/pen173.pdf">pen17.pdf</a>.</p>
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