document.write( "Question 1207618: I run a book club with $n$ people, not including myself. Every day, for $400$ days, I invite $2$ members in the club to review a book. What is the smallest positive integer $n$ so that I can avoid ever having the exact same group of $2$ members over all $400$ days? \n" ); document.write( "
Algebra.Com's Answer #845592 by ikleyn(52798)\"\" \"About 
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\n" ); document.write( "I run a book club with n people, not including myself. Every day, for 400 days,
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document.write( "Let the number of the members be n.\r\n" );
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document.write( "Then the number of pairs is  \"%28n%28n-1%29%29%2F2\".\r\n" );
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document.write( "They want you find the minimum possible n such that  \r\n" );
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document.write( "    \"%28n%2A%28n-1%29%29%2F2\" >= 400.    (1)\r\n" );
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document.write( "So, all you need is to solve this inequality.\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "    n*(n-1) >= 2*400 = 800.    (2)\r\n" );
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document.write( "Square root of 800 is  28.2842...\r\n" );
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document.write( "So, your number n is the first integer positive number \r\n" );
document.write( "which satisfies (2), and it is somewhere close to 28.\r\n" );
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document.write( "Check n= 28: 28*(28-1) = 28*27 = 756  <<<---=== not enough\r\n" );
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document.write( "Check n= 29: 29*(29-1) = 29*28 = 812  <<<---=== just enough\r\n" );
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document.write( "ANSWER.  You should have at least 29 members in the club to make it possible.\r\n" );
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