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Difference between revisions of "Transylvanian Hungarian MC (Romania) 2012 - G9 - P1"

(Created page with "Find all numbers <math>x,y\in\mathbb N</math> for which the relation <math> x+2y+\frac{3x}{y}=2012</math> holds. Proposed by Bela Kovacs")
 
 
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Proposed by Bela Kovacs
 
Proposed by Bela Kovacs
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Solution
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Solve for <math> x </math>.
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<math> x=-2y+2018-\frac{6054}{y+3} </math>
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Determine all value(s) of <math> y\in\mathbb{N} </math>
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<math> (y+3)|6054 </math> where the positive factors of <math> 6054 </math> are <math> 1, 2, 3, 6, 2018, 3027 </math> and <math> 6054 </math>.
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Equate each factor to <math> y+3 </math> and the values of <math> y </math> are <math> -1, 0, 3, 1006, 2015, 3024 </math> and <math> 6051 </math>, respectively.
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Find the value of <math> x </math> when
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<math> y= 3 </math> then <math> x= 1003 </math>
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<math> y=1006 </math> then <math> x=0 </math>
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<math> y= 2015 </math> then <math> x=-2015 </math>
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<math> y= 3024 </math> then <math> x=-4032 </math>
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<math> y= 6051 </math> then <math> x=-10085 </math>
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Answer: <math> x=1003 </math> and <math> y=3 </math>

Latest revision as of 03:18, 7 October 2016

Find all numbers $x,y\in\mathbb N$ for which the relation $x+2y+\frac{3x}{y}=2012$ holds.

Proposed by Bela Kovacs Solution Solve for $x$. $x=-2y+2018-\frac{6054}{y+3}$ Determine all value(s) of $y\in\mathbb{N}$ $(y+3)|6054$ where the positive factors of $6054$ are $1, 2, 3, 6, 2018, 3027$ and $6054$. Equate each factor to $y+3$ and the values of $y$ are $-1, 0, 3, 1006, 2015, 3024$ and $6051$, respectively. Find the value of $x$ when $y= 3$ then $x= 1003$ $y=1006$ then $x=0$ $y= 2015$ then $x=-2015$ $y= 3024$ then $x=-4032$ $y= 6051$ then $x=-10085$

Answer: $x=1003$ and $y=3$

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