Difference between revisions of "1988 OIM Problems/Problem 4"

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== Problem ==
 
== Problem ==
 
Let <math>ABC</math> be a triangle which sides are <math>a</math>, <math>b</math>, <math>c</math>.  We divide each side of the triangle in <math>n</math> equal segments.  Let <math>S</math> be the sum of the squares of the distances from each vertex to each of the points dividing the opposite side different from the vertices.
 
Let <math>ABC</math> be a triangle which sides are <math>a</math>, <math>b</math>, <math>c</math>.  We divide each side of the triangle in <math>n</math> equal segments.  Let <math>S</math> be the sum of the squares of the distances from each vertex to each of the points dividing the opposite side different from the vertices.
Prove that $\frac{S}{a^2+b^2+c^2} is rational.
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Prove that <math>\frac{S}{a^2+b^2+c^2}</math> is rational.
  
 
== Solution ==
 
== Solution ==
 
{{solution}}
 
{{solution}}

Revision as of 12:06, 13 December 2023

Problem

Let $ABC$ be a triangle which sides are $a$, $b$, $c$. We divide each side of the triangle in $n$ equal segments. Let $S$ be the sum of the squares of the distances from each vertex to each of the points dividing the opposite side different from the vertices. Prove that $\frac{S}{a^2+b^2+c^2}$ is rational.

Solution

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