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There is no such thing in the Federal tax law as 50/50, split, or joint custody.  The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year.  That parent is the custodial parent; the other parent is the noncustodial parent.)

Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the custody that a court might grant.).

The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody). That can usually only occur if both parents lived with the child at the same time.   And yes they are that picky.
See Custodial parent and noncustodial parent under the residency test in Pub 17
https://www.irs.gov/publications/p17/ch03.html - en_US_2015_publink1000170891)

Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household
-Earned Income Credit
-Child Care Credit

The non custodial parent can only claim: (Child would be listed as dependent)
-The Exemption
- The Child Tax Credit

See Special rule to divorced or separated parents (or parents who live apart).
https://www.irs.gov/publications/p17/ch03.html - en_US_2015_publink1000170897

But only if specifically specified in a pre-2009 divorce decree, separation agreement or the custodial spouse releases the exemption with a signed 8332 form - after 2009 the IRS only accepts a signed 8332 form that must be attached to the non-custodial parents tax return.