# Exercise 9.3.2

Analyze SELECT to show that if $n \ge 140$, then at least $\lceil n/4 \rceil$ elements are greater than the median-of-medians $x$ and at least $\lceil n/4 \rceil$ elements are less than $x$.

The problem can be reduced to the following inequality:

$$\frac{3n}{10} - 6 \ge \bigg\lceil \frac{n}{4} \bigg\rceil \\ \Downarrow \\ \frac{3n}{10} - 6 \ge \frac{n}{4} + 1 \\ \Downarrow \\ \frac{3n}{10} - 7 \ge \frac{n}{4} \\ \Downarrow \\ 12n - 280 \ge 10n \\ \Downarrow \\ 2n \ge 280 \\ \Downarrow \\ n \ge 140$$