Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it South Africa

If we place a $3$ at both ends of a number, its value is increased by $3372$ .
Find the original number.

Solution

Placing a $3$ at both ends of the number increases it by $3372$ , a four-digit number. Hence the original number is a two-digit number. Suppose the original number is $x$ . Then the number formed by placing a $3$ at both ends is equal to $3003 + 10x$ . This is $3372$ more than $x$ , that is,
$3003 + 10x - x = 3372 \implies 9x = 369 \implies x = 41.$

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