1990 AIME Problem 15

Find a_{}^{}x^5 + b_{}y^5 if the real numbers a_{}^{}, b_{}^{}, x_{}^{}, and y_{}^{} satisfy the equations 1&sid=ac36df6095c6f50b9a8349472df39d22">ax + by = 3^{}_{}, 1&sid=ac36df6095c6f50b9a8349472df39d22">ax^2 + by^2 = 7^{}_{}, 1&sid=ac36df6095c6f50b9a8349472df39d22">ax^3 + by^3 = 16^{}_{}, 1&sid=ac36df6095c6f50b9a8349472df39d22">ax^4 + by^4 = 42^{}_{}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.