Problem 1 - Entrance Test

To solve the equation 2^x + 5^x = 2 * 5^x, we first notice that 2 * 5^x can be written as 2 * 5^x = 10^x / 2^x. This observation leads us to rewrite the equation as 2^x + 5^x = 10^x / 2^x. Multiplying both sides of the equation by 2^x to clear the fraction gives us 2^(2x) + 2^x * 5^x = 10^x. Now, recognizing that 10^x = 2^x * 5^x, we substitute this into the equation, yielding 2^(2x) + 2^x * 5^x = 2^x * 5^x. Subtracting 2^x * 5^x from both sides, we get 2^(2x) = 0. This implies 2^x = 0 or, by the properties of exponents, x must be such that 2 raised to any power is not 0. This leads to the only feasible solution when considering the original equation's constraint that x must satisfy 2^x + 5^x = 2 * 5^x. Upon inspection, if x = 1, then 2^1 + 5^1 = 2 * 5^1, which simplifies to 2 + 5 = 2 * 5, and thus 7 = 10. Clearly, x = 1 is not a solution. This mistake in reasoning leads us to reconsider our steps. Let's approach it by trying to factor or simplify using properties of exponents directly applicable to the original equation. We should notice the possibility of x = 1 being a special case due to how the equation is structured. By testing x = 1 directly in the equation 2^x + 5^x = 2 * 5^x, we see 2^1 + 5^1 = 7 and 2 * 5^1 = 10, which does not satisfy the equation, indicating a mistake in assuming x = 1 could be a solution based on incorrect simplification. The error was in misinterpreting the result of the equation's manipulation. The correct path involves recognizing that if 2^x + 5^x = 2 * 5^x, then we can express this as 2^x = 2 * 5^x - 5^x = 5^x(2 - 1) = 5^x. This implies 2^x = 5^x. For this to be true, given the bases are different and both are greater than 1, x must make both sides equal. Considering positive integers, the smallest and most logical value to test is x = 1, but as seen, this does not yield equality. However, we are looking for a value of x that makes 2^x = 5^x, which suggests examining when the two exponential functions intersect. Since 2 < 5, 2^x grows slower than 5^x. The correct insight is to realize the question asks for a specific x where 2^x + 5^x equals 2 * 5^x, and the simplification or specific solution should directly address the original equation without overcomplicating it with incorrect substitutions or assumptions. The key is recognizing that 2^x = 5^x has a unique solution when considering real numbers, and that's x = 0 because 2^0 = 1 and 5^0 = 1. Substituting x = 0 into the original equation gives 2^0 + 5^0 = 1 + 1 = 2, and 2 * 5^0 = 2 * 1 = 2, thus validating x = 0 as a solution.