Find all pairs of consecutive **odd positive integers** both smaller than 10 such that their sum is more than 11. How many such pairs exist?

Let $x$ be the smaller of the two. Then $x + 2$ is the other, with $x \geq 1$ odd, $x + 2 < 10 \Rightarrow x < 8$, and $x + (x + 2) > 11 \Rightarrow x > 4.5$. Odd integers with $4.5 < x < 8$: $x \in \{5, 7\}$ — **2 values**. Pairs: $(5, 7)$ sum $12 > 11$ ✓; $(7, 9)$ sum $16 > 11$ ✓.