Let $\sum_{i=1}^n x_i \geqslant X$ and $\sum_{i=1}^n y_i \leqslant Y$ and $X, Y > 0$. Prove that exists such $i$ that $\frac{x_i}{y_i}\geqslant\frac{X}{Y}$.
I think that it could be proved using one of the following ways:
- pigeonhole principle,
- induction,
- by contradiction.
But I got stuck. For example when we try to show it by contradiction we can write that for all $i$ (if $\sum_{i=1}^n y_i$ is positive)
$$\frac{x_i}{y_i} < \frac{X}{Y}$$ $$\sum_{k=1}^n y_k\frac{x_i}{y_i} \leqslant Y\frac{x_i}{y_i} < X \leqslant \sum_{j=1}^n x_j $$ $$\sum_{k=1}^n y_k\frac{x_i}{y_i} < \sum_{j=1}^n x_j $$ $$\frac{x_i}{y_i} < \frac{\sum_{j=1}^n x_j}{\sum_{k=1}^n y_k}$$ But I don't know how it can be useful.
Do you have some hints?