Problem 1 - Entrance Test

To solve for x, we need to use numerical methods or observation. By trying the values, we can see that x = 3 satisfies the equation 3^3 + 7^3 = 10^3, since 27 + 343 = 370 and 1000 = 1000, which is close but not exact. However, when we try x = 3, we get 3^3 + 7^3 = 27 + 343 = 370 and 10^3 = 1000, which are different. We should notice, though, that as x increases, the term 10^x grows faster than 3^x + 7^x. Thus, there must be a solution for a value of x less than 3, but very close to it, because 3^x and 7^x are increasing, but 10^x increases faster. Looking at the given choices, we should realize the answer is less than 3 but should pick an integer for simplicity. Let's check x = 2: 3^2 + 7^2 = 9 + 49 = 58, and 10^2 = 100. Since 58 is less than 100, the value of x must be more than 2, but we already saw that it is less than 3. So, x must be between 2 and 3. However, since the given choices are integers, we should consider that the question might not be looking for an exact answer but rather the closest integer that could satisfy the condition given in the problem, or there might have been an error in the assessment of the provided solutions. Upon re-evaluation and trying x = 3 again with the proper method, we'd realize the actual task is beyond simple calculation and the exact value should be derived with logarithms or advanced numerical methods which typically aren't the focus for a quick AMC-style question. Given the options and looking back at the original task, without the possibility to apply more complex methods, we should look for a more suitable, less complex question that aligns with AMC style but recognize the oversight in calculation here.