Problems!

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

#Algebra #Geometry #Combinatorics #OlympiadMath #ProblemSolving

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Comments

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 - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$