Maths Olympiad Prep

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Number theory Difficulty 5.5 AIME, harder Find the answer

Problem 2. Alla thought of a three-digit number, in which there is no digit 0, and all digits are different. Bella wrote down the number in which the same digits are in reverse order. Galia subtracted the smaller number from the larger one. What digit stands in the tens place of the resulting difference?

#

A number or a short expression. Spacing and $ signs are ignored.

Solution

# Answer: 9

Solution. Since we are subtracting a smaller number from a larger one, the digit in the hundreds place of the first number is greater than that of the second. Then, in the units place, conversely, the digit of the first number is smaller than that of the second. Therefore, when subtracting, we will have to borrow a unit from the tens place. Initially, the same digits were in the tens place, but now the digit of the minuend is one less than that of the subtrahend. Therefore, we will have to borrow a unit from the hundreds place. In this case, in the tens place, the minuend ends up being 9 more than the subtrahend. Thus, the resulting digit is 9.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.