Question number 18, Ivana wants to write down five consecutive positive integers with the property that three of them have the same sum as the other two. How many different sets of five numbers can she write down? To begin, let's write down the sum of five positive integers in the simplest way that we can. So I will say that the first integer in our list is called M minus 2. The second one will be called M minus 1 and then so on. And the reason for doing this is that the expression for the sum is as simple as possible. And we have to make a note that M is at least 3 so that our first integer is at least 1. Okay. Then three of them will sum to a number that looks like 3M plus some remainder. And the other two will look like 2 times this number M plus possibly a different remainder. And the remainders are obtained by adding and subtracting 1s and 2s. So the smallest number we can construct in such a way is a minus 3. And the largest one would be a positive 3. Okay. Now with this, let's consider some additional properties. We know that 5M is the sum, but that's also what we get if we take the sum of the three first and then the sum of the two second. And we get a little relationship between A and B that says that A plus B is equal to 0. So A is equal to negative B. Next, we note that three of these numbers at a time have the same sum as the remaining two of them. So we have a relationship between M, A, and B that tells us that M is equal to 2A, for example. And so we are talking about an even number. We see that M is even, so let it be called 2 times an integer K. That integer, since M has to be at least 3, has to be at least 2. And let's go back and substitute that in to our equation we have here. I'll call that star. M is equal to 2A, but also equal to 2K. So A is equal to K. And from before, A has to be in the set we listed above. And A being equal to K has to be greater than or equal to 2. So A is equal to either 2 or A is equal to 3. And that gives us our final choices for M. We have two possibilities, M is equal to 4 or M is equal to 6. And from here, we can construct our list of positive consecutive integers. The first list will begin with a 2, and the second list will begin with a 4. So the middle numbers will be a 4 and a 6. And those are the only possibilities. We have constructed the answer, so there are no other possibilities. And the answer to number 18 is 2.

consecutive integers

integer sum

middle integer

even integer

integer sets