1. **Problem statement:** A baseball team plays at least one game each day for 30 days, with a total of no more than 45 games played in the month. We need to show that there exists a consecutive period of days during which exactly 14 games are played. 2. **Key idea:** Use the Pigeonhole Principle and consider the cumulative number of games played up to each day. 3. **Define cumulative sums:** Let $a_i$ be the number of games played on day $i$, where $1 \leq i \leq 30$. We know $a_i \geq 1$ and $\sum_{i=1}^{30} a_i \leq 45$. 4. Define the cumulative sums $S_k = \sum_{i=1}^k a_i$ for $k=1,2,\ldots,30$. Then: $$ 1 \leq S_1 < S_2 < \cdots < S_{30} \leq 45 $$ 5. Consider the set $\{S_1, S_2, \ldots, S_{30}\}$ and the set $\{S_1 + 14, S_2 + 14, \ldots, S_{30} + 14\}$. Both sets have 30 elements each. 6. All numbers in the first set are between 1 and 45, and all numbers in the second set are between 15 and 59. 7. Since there are 60 numbers total (30 in each set) but only 59 possible integers from 1 to 59, by the Pigeonhole Principle, there must be at least one number common to both sets. 8. Suppose $S_j = S_i + 14$ for some $i,j$ with $j > i$. Then the sum of games from day $i+1$ to day $j$ is: $$ S_j - S_i = 14 $$ 9. This means there is a consecutive period of days where exactly 14 games were played. **Final answer:** There must exist a consecutive period of days during which the team plays exactly 14 games.