A question on egyptian fractions An Egyptian fraction is the sum of distinct unit fractions. Are there any 2000 egyptian fractions that their sum is 1?

Yes! 
Because for any $k\geqslant 3$ we can write $1$ as sum of $k$ different fractions $\dfrac{1}{n}$. $\dfrac{1}{n}=\dfrac{1}{n+1}+\dfrac{1}{n(n+1)}$.
Apllying this identity we get: $1=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{42}=\dots$
