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 <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 13:07, 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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