1. **Problem statement:** Rasheed solves 5 coding challenges per competition, and Shayan solves 9 challenges per competition. They have the same total number of points after some competitions. We need to find the minimum number of competitions Rasheed participated in. 2. **Understanding the problem:** Let $r$ be the number of competitions Rasheed participated in, and $s$ be the number of competitions Shayan participated in. 3. **Formulating the equation:** Since they have the same total points, $$5r = 9s$$ 4. **Goal:** Find the smallest positive integer $r$ such that $5r$ is divisible by 9 (or equivalently, $5r$ equals $9s$ for some integer $s$). 5. **Solving the equation:** Rewrite as: $$5r = 9s \implies r = \frac{9s}{5}$$ For $r$ to be an integer, $9s$ must be divisible by 5. Since 9 and 5 are coprime, $s$ must be a multiple of 5. Let $s = 5k$ for some integer $k$. Then, $$r = \frac{9 \times 5k}{5} = 9k$$ 6. **Minimum positive integer solution:** Choose $k=1$: $$r = 9, \quad s = 5$$ 7. **Interpretation:** Rasheed must participate in at least 9 competitions for their total points to be equal. **Final answer:** $$\boxed{9}$$