finite solutions (CGMO2009/1)

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finite solutions (CGMO2009/1)

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Source: China Girls Mathematical Olympiad 2009, Problem 1

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#1 h Aug 18, 2009, 1:10 PM • 2 Y

Y by Adventure10, Mango247

Show that there are only finitely many triples $(x,y,z)$ of positive integers satisfying the equation $abc=2009(a+b+c).$

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#2 h Aug 18, 2009, 2:59 PM • 7 Y

Y by FlakeLCR, MathbugAOPS, Mathcollege, AntaraDey, Adventure10, Mango247, and 1 other user

In general, for positive integers $n$ and $k$ , prove that Diophantine equation $\prod_{i=1}^n x_i = k\sum_{i=1}^n x_i$ has a finite number of solutions made of positive integers.

Assume $1 \leq x_1 \leq x_2 \leq \cdots \leq x_n$ . Then
$\frac {1} {k} = \sum_{i=1}^{n} \frac {1} {\prod_{j \neq i} x_j} \leq \frac {n} {\prod_{i=1}^{n-1} x_i}, \ \textrm{ hence } \ \prod_{i=1}^{n-1} x_i \leq nk.$
Therefore $1 \leq x_i \leq nk$ , $1 \leq i \leq n-1$ . Since $x_n$ is uniquely determined, the conclusion ensues.

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#3 h Aug 20, 2009, 6:47 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

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#4 h Aug 20, 2009, 6:51 PM • 2 Y

Y by Adventure10, Mango247

orznorz, is that enough to prove that there are finitely many solutions?

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#5 h Aug 25, 2009, 6:19 AM • 2 Y

Y by Adventure10, Mango247

i think two answers above are both

right

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#8 h Apr 30, 2018, 12:05 PM • 2 Y

Y by Adventure10, Mango247

Let suppose that $a=max(a,b,c)$ . We have $abc=2009(a+b+c)<2009*3*a$ , so $bc<2009*3$ , which gives finite values of $b,c$ .If $bc<2009$ because $a<a+b+c$ , we have $abc<2009(a+b+c)$ , so $bc>2009$ . Because $abc=2009(a+b+c)$ , we obtain $a(bc-2009)=2009(b+c)$ .So $a=\frac{2009(b+c)}{bc-2009}$ .We know that $a$ is a positive integer so we chose only the values for $b,c$ such that $bc-2009 | 2009(b+c)$ but we know that these are finite. So we have a finite value for $a$ .

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#9 h Apr 30, 2025, 9:24 AM

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There was a similar problem dutch MO 2014 I posted my sol there if nybody wants he can take inspiration from that soln
disclaimer: that solution if applied in this one would use a lota of casework

This post has been edited 1 time. Last edited by Namisgood, May 3, 2025, 6:02 AM

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