Problem number 15. How many pairs of integers m and n satisfy the inequality absolute value of 2m-2023 plus absolute value of 2n-m is less than or equal to 1? Alright, so the absolute value is always non-negative, so it's 0 or greater. And since m and n are both integers, either the absolute value of 2m-2023 is 1, and then 2n-m is equal to 0, or the other way around. Because that's the only way. If we have one of them is equal to 2, then regardless of what the other one is equal to, since it's positive, that won't satisfy the inequality. So either one of them is equal to 1, or they're both equal to 0. Otherwise, they add up to more than 1, and the inequality is not satisfied. However, since 2 times m has a factor of 2, so it's even, and 2023 is odd, we have an even minus an odd number, so that must be equal to 1. And that means that 2n-m would be equal to 0, like we said earlier. So from 2m-2023 equals 1, we have two equations, either 2m-2023 equals 1, or negative 1. Solving each gives us m equals 1011, or 1012. Now we use that same logic again. Because 2n-m must equal 0, and 2n is even, m must also be even, so that we could have even minus even equals 0. So m must equal 1012. Plug that in, and we get that n equals 506. So there's only one valid pair of integers, m,n, that would satisfy this inequality, and that is 1012,506. So our answer is b, and we're done.

integer pairs

inequality

absolute expressions

even numbers

valid pair