Let $R$ be the rectangle on the cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$.

[i]Proposed by Boyan Litchev[/i]

[hide=Solution][i]Solution[/i]. $\boxed{85}$
We have $(6-n)(8-n)$ distinct squares with side length $n$, so the total number of squares is $5 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}$.[/hide]
To find the number of squares with sides parallel to the axes and vertices that are lattice points within the rectangle \( R \) with vertices \((0,0)\), \((5,0)\), \((5,7)\), and \((0,7)\), we need to count the number of such squares for each possible side length \( n \).

1. **Identify the dimensions of the rectangle:**
   - The rectangle \( R \) has a width of 5 units and a height of 7 units.

2. **Count the number of squares for each side length \( n \):**
   - For a square of side length \( n \), the top-left vertex of the square can be placed at any lattice point \((x, y)\) such that \(0 \leq x \leq 5 - n\) and \(0 \leq y \leq 7 - n\).
   - Therefore, the number of such squares with side length \( n \) is \((6 - n)(8 - n)\).

3. **Sum the number of squares for all possible side lengths \( n \):**
   - The possible side lengths \( n \) range from 1 to the minimum of the width and height of the rectangle, which is 5.
   - We calculate the number of squares for each \( n \) and sum them up:
     \[
     \begin{aligned}
     &\text{For } n = 1: (6 - 1)(8 - 1) = 5 \cdot 7 = 35, \\
     &\text{For } n = 2: (6 - 2)(8 - 2) = 4 \cdot 6 = 24, \\
     &\text{For } n = 3: (6 - 3)(8 - 3) = 3 \cdot 5 = 15, \\
     &\text{For } n = 4: (6 - 4)(8 - 4) = 2 \cdot 4 = 8, \\
     &\text{For } n = 5: (6 - 5)(8 - 5) = 1 \cdot 3 = 3.
     \end{aligned}
     \]

4. **Add the results:**
   \[
   35 + 24 + 15 + 8 + 3 = 85.
   \]

Conclusion:
The total number of squares with sides parallel to the axes and vertices that are lattice points within the region bounded by \( R \) is \(\boxed{85}\).