1988 OIM Problems/Problem 4: Difference between revisions
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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 <math>\frac{S}{a^2+b^2+c^2}</math> is rational. | Prove that <math>\frac{S}{a^2+b^2+c^2}</math> is rational. | ||
== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
Revision as of 12:07, 13 December 2023
Problem
Let 
be a triangle which sides are 
, 
, 
. We divide each side of the triangle in 
equal segments. Let 
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 
is rational.
Solution
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