Arduino DUE does not start until is not pressed Reset button

If you have an Arduino older than Uno Rev3, you may be suffering from the "reset overshoot" problem described here:
Regression between uno and uno R2 VALIDATED. HARDWARE PROBLEM CONFIRMED - Suggestions for the Arduino Project - Arduino Forum These were especially common in the Uno R2.
The fix is a clamping diode (any general purpose diode) between reset and +5V (cathode on +5V, anode on reset), which you can do like: