2020 USAMTS Round 1 Problems/Problem 5
Find all pairs of rational numbers such that and .
Solution 1
We claim that there is only one solution, which is the solution pair .
Let , where is a rational number greater than . Then when we substitute that into the equation , we get . We want to separate the variables from each other, so we can multiply by to get . Moving to the other side, and using the exponent rule, we get . Multiplying this by , we get . Substituting that into , we get , so . Therefore our solution set is , but wait! The solution set contains irrational numbers, and the question only asks for rational numbers. Therefore, and both need to be integers. By inspection, we can see that the only solution is , giving us the solution set .
Solution and by smartguy888
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