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Score Secrets


Farhan shows his test score to Muaz, Bristy, and Mursalin, but everyone else keeps it hidden. Muaz thinks, “At least two of us get the same score.” Bristy thinks, “I didn’t get the lowest score.” Mursalin thinks, “I didn’t get the highest score.”

  • Who got the highest marks? 
  • Who got the lowest marks?


Source: BdMO 2019 National Junior P1


Proof Based Problems  


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Solution

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Bristy got the highest marks and Mursalin got the lowest marks.


Let $B,F,Z,S$ be the scores of Bristy, Farhan, Muaz and Mursalin respectively.


Muaz determined that two of them got the same score by only knowing his own and Farhan's score, so $F=Z$.

Bristy determined that she didn't get the lowest score by only seeing Farhan's score, so $F<B$.

Mursalin determined that he didn't get the highest score by only seeing Farhan's score, so $F>S$.


\[\therefore B>F=Z>S\]


So, Bristy got the highest marks and Mursalin got the lowest marks.

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