4. (7 points) In a bag, there are 15 balls (see the figure). Color each ball in one of three colors: blue, green, or red - so that two of the statements are true, and one is false:

- there is one more blue ball than red balls;
- there are an equal number of red and green balls;
- there are 5 more blue balls than green balls.

## Write in detail how you reasoned.

![](https://cdn.mathpix.com/cropped/2024_05_06_ddb3d7b713c1942072ecg-2.jpg?height=425&width=1194&top_left_y=1501&top_left_x=434)
# Answer.

![](https://cdn.mathpix.com/cropped/2024_05_06_ddb3d7b713c1942072ecg-3.jpg?height=460&width=1246&top_left_y=327&top_left_x=405)

7 blue balls, 6 red balls, 2 green balls.

Solution. We will prove that the second statement cannot be true. Indeed, if the first and second statements are true, then if we remove one blue ball, the number of balls of each color should be equal.

But \(15-1=14\) balls cannot be evenly divided into 3 colors. Now, let's assume the second and third statements are true. Then if we remove 5 blue balls, the number of balls of each color should again be equal.

But \(15-5=10\) balls cannot be evenly divided into 3 colors.

Thus, only the first and third statements can be true.

We can reason in different ways from here.

First method. If we add 1 red ball to the bag, the number of blue and red balls will be equal, and if we add 5 more green balls, the number of balls of each color will be the same, specifically, there will be \((15+1+5):3=7\) balls of each color.

Now we can count how many balls of each color were in the bag: 7 blue balls, \(7-1=6\) red balls, and \(7-5=2\) green balls.

Second method. From the true statements 1 and 3, it follows that there are 4 fewer green balls than red balls. If we remove 5 blue balls and 4 red balls from the bag, the number of balls of each color will be the same, specifically, there will be \((15-5-4):3=2\) balls of each color. Thus, there were 2 green, 6 red, and 7 blue balls in the bag.

It is also possible to solve the problem using an equation.

## Grading Criteria.

- Any correct and complete solution (the correct statements are chosen, the number of balls of each color is calculated, and explanations are provided) 7 points.
- The correct statements are indicated but not justified, and the number of balls of each color is correctly found based on this - 4 points.
- The correct statements are justified, but the number of balls of each color is not found or is found incorrectly - 3 points.
- An incomplete enumeration of color options is performed, and the correct answer is found - 1 point.
- Only the answer is provided - 0 points.