For many-electron atoms, spherically averaged electron-electron coalescence h(0)(R) and counterbalance d(0)(u) functions are studied which, respectively, represent the probability densities that any electron pairs with zero interelectronic distance are located at a radius R from the nucleus and that any electron pairs with zero center-of-mass radius have a relative distance u. For the exact and Hartree-Fock (HF) wave functions, cusp relations h(0)'(0)/h(0)(0) = -4Z and d(0)'(0)/d(0)(0) = -2Z are derived theoretically, where the prime denotes the first derivative and Z is nuclear charge. At the Hartree-Fock limit level, both functions h(0)(R) and d(0)(u) are found to be monotonically decreasing with a single maximum at R = 0 or u = 0 for all the 102 atoms He through Lr. The long-range asymptotic behavior of the coalescence and counterbalance functions is governed in general by the orbital energy of the highest occupied atomic orbital.