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Problems!

  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
3 years, 5 months ago

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Solution to Problem 1

\(\frac{2RcosA}{a}=\frac{2RcosB}{b}=\frac{2RcosC}{c}\)

By cosine rule,

\(\frac{b^2+c^2-a^2}{2abc}=\frac{c^2+a^2-b^2}{2abc}=\frac{a^2+b^2-c^2}{2abc}\)

Hence \(a=b=c\)

Souryajit Roy - 3 years, 5 months ago

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