 Vereitinov akakwanisa kupinda mumakwikwi maviri ekushambira kumaOlympics ezhizha ra2004 kuAthens, nekubvisa FINA B-cut ye1:52.55 200 m freestyle kubva kuEuropean Championships kuMadrid, Spain. Mumakwikwi emamita 200 akasununguka, Vereitinov akavhara zvakanakisisa zvake zve1:51.38 kuti atungamirire kupisa kwechitatu, asi akapedza chete munzvimbo makumi maviri nemanomwe kubva mumatangiro emangwanani. Akashandawo pamwe naSerhiy Fesenko, Serhiy Advena, naMaxim Kokosha mu 4 × 200 m freestyle relay. Achishambira muchikamu chechitatu mumwaka wekutanga, Vereitinov akarekodha kupatsanurwa kwe1:50.34, asi maUkraine akagadzikana nenzvimbo yechitanhatu uye yegumi nembiri yakazara, munguva yekupedzisira ye7:24.13. 