Track / Stage 4 / 140 of 340 #400 of 2000

Problem 400

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

4. 2005 cells are arranged in a row, to be painted with red, white, and black colors, with each cell painted one color, and it is required that the number of cells painted red must be even. Then there are $\qquad$ different ways to paint them.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。

Note: The $\qquad$ is kept as is, since it likely represents a space for the answer to be filled in.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

4. $\frac{1}{2}\left(3^{2005}+1\right)$ Hint: Based on the conclusion of Example 7.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.