We obtain an explicit expression for the center-of-mass ( CM ) energy of two colliding general geodesic massive and massless particles at any spacetime point around a Kerr black hole . Applying this , we show that the CM energy can be arbitrarily high only in the limit to the horizon and then derive a formula for the CM energy of two general geodesic particles colliding near the horizon in terms of the conserved quantities of each particle and the polar angle . We present the necessary and sufficient condition for the CM energy to be arbitrarily high in terms of the conserved quantities of each particle . To have an arbitrarily high CM energy , the angular momentum of either of the two particles must be fine-tuned to the critical value L _ { i } = \Omega _ { H } ^ { -1 } E _ { i } , where \Omega _ { H } is the angular velocity of the horizon and E _ { i } and L _ { i } are the energy and angular momentum of particle i ( = 1 , 2 ) , respectively . We show that , in the direct collision scenario , the collision with an arbitrarily high CM energy can occur near the horizon of maximally rotating black holes not only at the equator but also on a belt centered at the equator . This belt lies between latitudes \pm \mbox { acos } ( \sqrt { 3 } -1 ) \simeq \pm 42.94 ^ { \circ } . This is also true in the scenario through the collision of a last stable orbit particle .