(at around 1h 28 mins) When Alex is shot in the arm, the police officer's slide on his pistol locks in the rear position indicating his gun is out of ammunition though he fired a single shot. This means the cop responded to a 911 emergency with one bullet in his gun.
(at around 29 mins) When Alex visits the artist's apartment the first time for the chess rematch, the timer's button is in a different position when they sit down than when they actually start the game.
(at around 1h 22 mins) When Alex discovers his dead brother, he throws up in the sink then stumbles over the body, and picks up a blue bottle containing alcohol. In the following scenes, that same bottle reappears then disappears in the spot where Alex moved it from.
(at around 28 mins) At the main character's first visit to the "artist's apartment", he stands and looks at a painting (the one of the naked woman lying on the couch). He stands still to look at it and, as the camera shifts from right to left around him, the TV in the background (which is turned off), picks up the legs of someone walking, yet both characters are standing perfectly still.
(at around 40 mins) When Alex is in the ambulance, the paramedic is incorrectly wearing his phonendoscope (the eartips should point forward). Moreover, it's quite difficult to hear chest sounds through many clothing layers (as shown in the movie) so the diaphragm's usually placed directly on the skin.