document.write( "Question 487277: Пlease help me with those questions!!!
\n" ); document.write( "1. Find all triples (x; y; z) of positive integers such that x < y < z and
\n" ); document.write( "(1/x)-(1/xy)-(1/xyz)=19/9
\n" ); document.write( "2. Show that from any five integers, one can always choose three of these integers such that their sum is divisible by 3.
\n" ); document.write( "Thank you very much!!!
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Algebra.Com's Answer #333084 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
1. There are none. Since x < y < z, the maximum value of the LHS is about 1 (set x = 1, y,z to be large numbers). The LHS can never get larger than 1, so there are no solutions (x,y,z) of positive integers.\r
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\n" ); document.write( "\n" ); document.write( "2. Out of any five integers, there will either exist an integer for each residue modulo 3 (in which we're done), or all five integers will occupy two different residues mod 3. If this is the case, then by a simple Pigeonhole argument, three of them have the same residue mod 3, in which we're also done (since the sum of these three is 0 mod 3). \n" ); document.write( "

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