Methotraxate is one of the drugs often used in chemotherapy. It interferes with folate metabolism by acting as a competitive inhibitor of the enzyme dihydrofolate reductase. Without this enzyme, the cells are starved for tetrahydrofolate, and unable to make purines (adenine, guanine) and dTPM. This is especially toxic to fast-growing cells, like cancer cells.
How does a cancer become resistant to this drug? The cells increase the expression of the gene that causes manufacture of dihydrofolate reductase. With an increased volume of the enzyme hanging around, the inhibition of the enzyme by methotrexate is overcome at normal doses, and the tumor becomes "resistant" to the drug.