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Italian Mathematical Olympiad

Algebra Difficulty 4.7 AIME Find the answer Italy

Problem:

In a bag there are marbles of various colors. It is known that all the marbles except 6 are yellow, all the marbles except 7 are red, all the marbles except 10 are blue. Moreover, there is at least one blue marble and there could also be marbles of colors different from yellow, red and blue. How many marbles does the bag contain?

Pick one

Solution

Solution:

The answer is (B). Let us call $n$ the total number of marbles present in the bag. The yellow marbles are then $n-6$ , the red ones are $n-7$ , and the blue ones are $n-10$ . Since there could also be marbles of other colors, the following inequality holds: $n-6+n-7+n-10 \leq n$ , which implies $n \leq \frac{23}{2}$ . Since there is at least one blue marble, $n \geq 11$ , from which $n=11$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.