Can you help with the problem: [a] is the largest integer part of the number a, not exceeding it. Solve the equation: [x]+[2x]=2018?

Answer from: Svetlana Zh.:
5,7 K...

First, [a] is not the "the largest integer part". It is simply an integer part, which is defined as the largest integer not exceeding a. For example [1] = 1, [1.5] = 1, [1.999] = 1, [2] = 2. Now to the solution.
First, let's solve an easier equation : x + 2x = 2018. its solution x = 672.666.... it is clear that the solutions of the original equation cannot be very different from this one, because [x] differs from x by no more than 1. Next, let's see at which points, as x increases, the left-hand side will change value. it is obvious that [x] will experience a jump of 1 at integer points of x, and [2x] at integer and integer and a half. Let us check some such values near the solution of the auxiliary equation.
[672] + [2*672] = 3*672 = 2016
[672.5] + [2*672.5] = 672 + 1345 = 2017
[673] + [2*673] = 3*673 = 2019
alas, the left-hand side of the equation has a jump of 2 at integer points, and the 2018 value just jumped. The function is obviously monotonically increasing, so there are no solutions

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