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    OlympiadMediumNumber Theory5–6

    How many two-digit numbers have digits that add up to 1010?

    Quantos números de dois algarismos têm algarismos cuja soma é 1010?

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    Solution

    The tens digit can be 1,2,…,91, 2, \ldots, 9 and the units digit is then fixed: 19,28,37,46,55,64,73,82,9119, 28, 37, 46, 55, 64, 73, 82, 91. A tens digit of 00 is not a two-digit number, so there are 99.