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Print75th Romanian Mathematical Olympiad
Romania algebra
Problem
Alexia has several marbles and her friend, Cristina, has none. Each day of one week, starting on Monday, Alexia gave Cristina some of her marbles, so that each day Alexia gave more marbles than the day before. On Monday Alexia gave five times less marbles than on Friday, on Tuesday she gave six times less marbles than on Saturday, and on Wednesday she gave seven times less marbles than on Sunday. At the end of that week, Cristina got 72 marbles. Find how many marbles got Cristina Thursday. Lucian Dragomir
Solution
Denote the number of marbles given by Alexia from Monday until Sunday, where and . Then , , , hence (1).
If , then , contradicting (1). Therefore, or .
If , then and yields , , , whence , contradicting (1). So, .
From follows . Moreover, (1) gives (2).
If , then , contradicting (1), hence . From follows .
Now (2) gives , hence , so . But , hence , and yields .
Clearly is acceptable (the number of the given marbles being 2, 3, 4, 7, 10, 18, 28). So, the answer is 7 marbles.
If , then , contradicting (1). Therefore, or .
If , then and yields , , , whence , contradicting (1). So, .
From follows . Moreover, (1) gives (2).
If , then , contradicting (1), hence . From follows .
Now (2) gives , hence , so . But , hence , and yields .
Clearly is acceptable (the number of the given marbles being 2, 3, 4, 7, 10, 18, 28). So, the answer is 7 marbles.
Final answer
7
Techniques
IntegersTechniques: modulo, size analysis, order analysis, inequalitiesLinear and quadratic inequalities