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  1. Let \(H\) be the orthocenter of triangle \(ABC\). Prove that, if \(\dfrac{AH}{BC}=\dfrac{BH}{CA}=\dfrac{CH}{AB}\), the triangle is equilateral.

  2. Let \(a, b, c\) be the roots of \(x^3-x^2-x-1\). Prove that \(\dfrac{a^{2014}-b^{2014}}{a-b}+\dfrac{b^{2014}-c^{2014}}{b-c}+\dfrac{c^{2014}-a^{2014}}{c-a}\) is an integer.

  3. Let \(A\) and \(B\) be two subsets of \(S=\{1,...,2000\}\) such that \(|A|\cdot|B|\ge3999\). For a set \(X\), let \(X-X\) denote the set \(\{x-y|x,y\in{X}\}\). Prove that \((A-A)\cap(B-B)\) is not an empty set.

Note by José Marín Guzmán
2 years, 3 months ago

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Solution to Problem 1


By cosine rule,


Hence \(a=b=c\) Souryajit Roy · 2 years, 3 months ago

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