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UKMT Senior Kangaroo, 2021

Number theory Difficulty 3.0 AMC 10/12 Prove it United Kingdom

Five cards have the numbers 101, 102, 103, 104 and 105 on their fronts. On the reverse, each card has one of five different positive integers: $a$ , $b$ , $c$ , $d$ and $e$ respectively. We know that $c = be$ , $a + b = d$ and $e - d = a$ . Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?

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