[SalesForce] Passing Picklist Values to Controller with InputField in VisualForce

I'm new to VisualForce and trying to figure out how to pass the values a user selected from a birthday picklist into my controller. The picklists are fields of my Contact object and one is dependent on the other (the month selected determines the days selectable). Whenever I try to submit a value, nothing shows up (I have at least one contact for every birthday). How do I do this correctly?

Here is my VisualForce code:

<apex:page controller="BirthdayController">
    <apex:form >
        <h3>Enter Your Birthday:</h3>
        Month: <apex:inputField value="{!c.Month__c}"/>
        Day: <apex:inputField value="{!c.Day__c}"/>
        <apex:commandButton value="Submit" action="{!submit}"/>
        <apex:pageBlock title="Birthdays">
            <apex:pageBlockTable value="{!Birthdays}" var="b">
                <apex:column value="{!b.FirstName}"/>
                <apex:column value="{!b.LastName}"/>
            </apex:pageBlockTable>
        </apex:pageBlock>
    </apex:form>
</apex:page>

And here's my controller code:

public class BirthdayController
{
    public Contact c {get; set;}
    public String bday;
    public void submit()
    {
        bday = c.Month__c + '/' + c.Day__c;
    }
    public List<Contact> getBirthdays()
    {
        List<Contact> results = [SELECT FirstName, LastName FROM Contact WHERE Birthday__C = :bday];
        return results;
    }
}

EDIT: Basically what I'm trying to do is make it so that a user can select a date using the picklists and have all the contacts with matching birthdays show up. I know I could do this by manually creating a populating a selection list, but I want to use the contact fields because they are dependent. Is what I'm trying to do possible?