Question 180
On a line, 200 points are marked and numbered $1, 2, 3, \ldots, 200$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $200$. In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for every pair $(i, j)$ with $1 \le i \lt j \le 200$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two. Show that the number of crickets was at least $10000$ and that there is a way that $10000$ crickets could jump satisfying the conditions above.