Coin Conundrum / Puzzle

There's no solution unless there's a coin denomination other than $1, $0.25, $0.10, $0.05 and $0.01.

Here's proof.

• Observation 1: $1 coin is not one of the 6 coins. Because adding up to exactly $0.15 from 5 coins is impossible.
  
• Observation 2: You can have at most 1 $0.50 coin. Because if you had more than 1 $0.50 coin then you could change for $1 bill.
  
• Observation 3: You can have at most 1 $0.25 coin. Because if you had more than 1 $0.25 coin then you could change for $0.5.
  
  • Theory A: One of the 6 coins is a $0.50 coin.
    Implication: The other 5 coins add up to exactly $0.65.
    
    • Theory a: One of the 5 coins is a $0.25 coin.
      Implication: The other 4 coins add up to exactly $0.40.
      
      Then the 4 remaining coins must be all $0.10 coins. Because if they were a lesser demonination they would add up to less than $0.40.
      
      However this combination can produce exact change for $0.95 with 1 $0.50, 1 $0.25 and 2 $0.10 and therefore is wrong.
      
      Conclusion a: None of the 5 coins is a $0.25 coin.
      
      Implication: The highest possible denomination for the 5 coins is $0.10. However 5 * $0.10 < $0.65.
      
    Conclusion A: None of the 6 coins is a $0.50 coin.
    
    
  • Theory B: One of the 6 coins is a $0.25 coin.
    Implication: The other 5 coins add up to exactly $0.90. Since there cannot be more than 1 $0.25, the highest possible denomination of these 5 coins is $0.10.
    
    However 5 * $0.10 < $0.90.
    
    Conclusion B: None of the 6 coins is a $0.25 coin.
    
  
  Therefore the highest possible denomination for the 6 coins is $0.10.
  
  However 6 * $0.10 < $1.15.
  

Edit: damnit ninja