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2024 AMC 12A Problem 3

Problem 3 of 25EasierAlgebra

The number 20242024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

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Solution

Each number is at most 99,99, so kk numbers sum to at most 99k.99k. Since 9920=1980<2024,99\cdot20=1980\lt2024, at least 2121 numbers are required. With 21,21, we can use twenty 9999s and one 4444: 2099+44=1980+4420\cdot99+44=1980+44 =2024.=2024. Thus, the correct answer is B.

More practice

Concepts: optimization · bounding to limit cases

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.