do while loop not working properly

Question

I want the user to input a date that falls before or equal to May 30,2016 and not below than January 1,2016. If the input is not valid then it should go back and ask question again to the user. The problem is that do...while loop is not looping, it stops after the input is invalid. Here's the code:

#include<iostream>
using namespace std;

int main(){
    int month,day,year;
    char symbol;

    do{
        cout<<"Enter date: ";
        cin>>month>>symbol>>day>>symbol>>year;

        if(symbol=='/'&&month<=05&&day<=31&&year==2016){
            cout<<month<<symbol<<day<<symbol<<year;

            if(month==01){          // this code is for month of January
                if(day==3||day==10||day==17||day==24||day==31){
                    cout<<" is Sunday.";
                }
                else if(day==4||day==11||day==18||day==25){
                    cout<<" is Monday.";
                }
                else if(day==5||day==12||day==19||day==26){
                    cout<<" is Tuesday.";
                }
                else if(day==6||day==13||day==20||day==27){
                    cout<<" is Wednesday.";
                }
                else if(day==7||day==14||day==21||day==28){
                    cout<<" is Thursday.";
                }
                else if(day==1||day==8||day==15||day==22||day==29){
                    cout<<" is Friday.";
                }
                else if(day==2||day==9||day==16||day==23||day==30){
                    cout<<" is Saturday.";
                }

            }
            else if(month==02){     //this code is for month of February
                if(day==7||day==14||day==21||day==28){
                    cout<<" is Sunday.";
                }
                else if(day==1||day==8||day==15||day==22||day==29){
                    cout<<" is Monday.";
                }
                else if(day==2||day==9||day==16||day==23){
                    cout<<" is Tuesday.";
                }
                else if(day==3||day==10||day==17||day==24){
                    cout<<" is Wednesday.";
                }
                else if(day==4||day==11||day==18||day==25){
                    cout<<" is Thursday.";
                }
                else if(day==5||day==12||day==19||day==26){
                    cout<<" is Friday.";
                }
                else if(day==6||day==13||day==20||day==27){
                    cout<<" is Saturday.";
                }   
            }
            else if(month==03){         //this code is for month of March

                if(day==6||day==13||day==20||day==27){
                    cout<<" is Sunday.";
                }
                else if(day==7||day==14||day==21||day==28){
                    cout<<" is Monday.";
                }
                else if(day==1||day==18||day==15||day==22||day==29){
                    cout<<" is Tuesday.";
                }
                else if(day==2||day==9||day==16||day==23||day==30){
                    cout<<" is Wednesday.";
                }
                else if(day==3||day==10||day==17||day==24||day==31){
                    cout<<" is Thursday.";
                }
                else if(day==4||day==11||day==18||day==25){
                    cout<<" is Friday.";
                }
                else if(day==5||day==12||day==19||day==26){
                    cout<<" is Saturday.";
                }
            }
            else if(month==04){     //this code is for month of April

                if(day==3||day==10||day==17||day==24){
                    cout<<" is Sunday.";
                }
                else if(day==4||day==11||day==18||day==25){
                    cout<<" is Monday.";
                }
                else if(day==5||day==12||day==19||day==26){
                    cout<<" is Tuesday.";
                }
                else if(day==6||day==13||day==20||day==27){
                    cout<<" is Wednesday.";
                }
                else if(day==7||day==14||day==21||day==28){
                    cout<<" is Thursday.";
                }
                else if(day==1||day==8||day==15||day==22||day==29){
                    cout<<" is Friday.";
                }
                else if(day==2||day==9||day==16||day==23||day==30){
                    cout<<" is Saturday.";
                }
            }   
            else if(month==05){     //this code is for month of May

                if(day==1||day==8||day==15|day==22||day==29){
                    cout<<" is Sunday.";
                }
                else if(day==2||day==9||day==16||day==23||day==30){
                    cout<<" is Monday.";
                }
                else if(day==3||day==10||day==17||day==24||day==31){
                    cout<<" is Tuesday.";
                }
                else if(day==4||day==11||day==18||day==25){
                    cout<<" is Wednesday.";
                }
                else if(day==5||day==12||day==19||day==26){
                    cout<<" is Thursday.";
                }
                else if(day==6||day==13||day==20||day==27){
                    cout<<" is Friday.";
                }
                else if(day==7||day==14||day==21||day==28){
                    cout<<" is Saturday.";
                }
            }
            break;  
        }
        else{
            cout<<"You have entered an invalid input.\n"<<endl;
        }
    }while(symbol!='/'&&!(month<=05)&&!(day<=31)&&year!=2016);
    return 0;
}

Answer 1

while(symbol!='/'&&!(month<=05)&&!(day<=31)&&year!=2016);

must be

while(symbol!='/' || !(month<=05) || !(day<=31) || year!=2016);

or as suggested by @Barmar

while(symbol!='/' || (month>05) || (day>31) || year!=2016);

Your original expression requires all 4 to be true due to the && . The expression using ||requires just one of them to be true.

Also notice that you don't need that check as you already have an if inside.

So you could just do:

while (true) {
    cout<<"Enter date: ";

    cin>>month>>symbol>>day>>symbol>>year;

    // clear cin
    cin.clear();
    cin.ignore(10000,'\n');  // Or better:
                             // std::numeric_limits<std::streamsize>::max()
                             // instead of just 10000 

    if(symbol=='/' && month<=05 && day<=31 && year==2016){
        // code for handling valid date here

        // This will end the while loop
        break;  
    }
    else{
        cout<<"You have entered an invalid input.\n"<<endl;
    }
};

Unrelated: Also notice this line

cin>>month>>symbol>>day>>symbol>>year;
            ^^           ^^

Here you read into symbol twice, ie the second value will just overwrite the first. Instead you would need

cin >> month >> symbol1 >> day >> symbol2 >> year;
                      ^                 ^

and of cause change the rest of the code accordingly.

Update

The code can be tested here https://ideone.com/KvCy1e

With the input

dem/dem/dem
1/1/2016

the output is

Enter date: You have entered an invalid input.

Enter date: 1/1/2016 is Friday.

Update 2

As noticed by @Barmar in a comment, you should initialize your variables to some invalid date before the loop.

int month = 13;
int day = 32;
int year = 0;
char symbol = 'x';

And btw: Since you use ints you should check for negative values.