Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$.

[i]Proposed by Evan Chen[/i]
1. **Claim**: The smallest positive integer \( c \) such that for any positive integer \( n \), the decimal representation of the number \( c^n + 2014 \) has digits all less than 5 is \( c = 10 \).

2. **Verification for \( c = 10 \)**:
   - For any positive integer \( n \), \( 10^n \) is a power of 10, which means it has a 1 followed by \( n \) zeros.
   - Adding 2014 to \( 10^n \) results in a number of the form \( 1000\ldots 02014 \) (with \( n-1 \) zeros between the 1 and 2).
   - Clearly, all digits in \( 10^n + 2014 \) are less than 5, as the digits are 1, 0, 2, 0, 1, and 4.

3. **Proof for \( c < 10 \)**:
   - We need to show that for any \( k < 10 \), there exists some positive integer \( n \) such that \( k^n + 2014 \) has a digit greater than or equal to 5.

4. **Case \( k = 1 \) to \( 5 \)**:
   - For \( k = 1 \):
     \[
     1^n + 2014 = 1 + 2014 = 2015 \quad (\text{for } n = 1)
     \]
     The digit 5 appears in the number 2015.
   - For \( k = 2 \):
     \[
     2^n + 2014 \quad (\text{for } n = 1)
     \]
     \[
     2 + 2014 = 2016
     \]
     The digit 6 appears in the number 2016.
   - For \( k = 3 \):
     \[
     3^n + 2014 \quad (\text{for } n = 1)
     \]
     \[
     3 + 2014 = 2017
     \]
     The digit 7 appears in the number 2017.
   - For \( k = 4 \):
     \[
     4^n + 2014 \quad (\text{for } n = 1)
     \]
     \[
     4 + 2014 = 2018
     \]
     The digit 8 appears in the number 2018.
   - For \( k = 5 \):
     \[
     5^n + 2014 \quad (\text{for } n = 1)
     \]
     \[
     5 + 2014 = 2019
     \]
     The digit 9 appears in the number 2019.

5. **Case \( k = 6 \) to \( 9 \)**:
   - For \( k = 6 \):
     \[
     6^n + 2014 \quad (\text{for } n = 2)
     \]
     \[
     6^2 + 2014 = 36 + 2014 = 2050
     \]
     The digit 5 appears in the number 2050.
   - For \( k = 7 \):
     \[
     7^n + 2014 \quad (\text{for } n = 2)
     \]
     \[
     7^2 + 2014 = 49 + 2014 = 2063
     \]
     The digit 6 appears in the number 2063.
   - For \( k = 8 \):
     \[
     8^n + 2014 \quad (\text{for } n = 2)
     \]
     \[
     8^2 + 2014 = 64 + 2014 = 2078
     \]
     The digit 7 appears in the number 2078.
   - For \( k = 9 \):
     \[
     9^n + 2014 \quad (\text{for } n = 2)
     \]
     \[
     9^2 + 2014 = 81 + 2014 = 2095
     \]
     The digit 9 appears in the number 2095.

6. **Conclusion**:
   - For \( c = 10 \), \( 10^n + 2014 \) always has digits less than 5.
   - For any \( c < 10 \), there exists some \( n \) such that \( c^n + 2014 \) has a digit greater than or equal to 5.

Therefore, the smallest positive integer \( c \) is \( 10 \).

The final answer is \( \boxed{10} \)