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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 $$a=b=c$$

- 3 years, 5 months ago