Question number 25. The numbers from 1 to 10 are to be written around a circle in some order. Then each number will be added to its two immediate neighbors to obtain a list of 10 new numbers. What is the largest possible value of the smallest number in the new list? So let us begin by making an example so we can better understand the question. I will arrange the numbers from 1 to 10 in the following way. 3 followed by 5, followed by 7, 8, 1, 6, 10, 2, 4, 9, and then we'll have the repetition because they're around a circle. So that's my list. And then I will take three numbers at a time to make the new list. So 3 is replaced by the sum of 9, 3, and 5. So that gives me 17. And then 5 is replaced with the sum 3 plus 5 plus 7. So that gives me 15. And then I can keep going like that and make a new list. And so on. Then they start to repeat. So again, 3 would be replaced by 9 plus 3 plus 5. That gives us 17 and so forth. So that's what the problem is requiring us to do. And what we have to decide on is what is the largest possible value of the smallest number in the list? In our new list, the number that is the smallest is 15. That appears three times. And so the question is what is the largest possible value? Can we have something that's larger than 15? So let us try to come up here with an upper bound. And if I take any three numbers here in the new list, let's say I take the smallest three I can find in a row. So I guess that is 16, 15, and 16. And the smallest sum I can obtain, the least sum possible would be the sum of three numbers, x, y, and z, let's say. But without any overlap here, without considering the neighbors of each of those numbers, remember 16 came from 2 and the neighbors, 4 resulted in 15 when we added the neighbors together. And so like that, I don't want any overlap. I just sum the smallest numbers I have one at a time without considering any of them twice. And so that gives me 45, the first nine positive integers. And if I require that that 45 is also the sum of three numbers, then what we see is that not all of the x, y, z can be greater than 15. And so that's how we come up with our upper bound. So 15 is an upper bound on our number. That's the greatest possible value of the least number. But then by means of our example here, the one I just wrote above, that is in fact a possibility. So by means of an example, we have produced such a thing. And so we conclude that 15 is the value that we are looking for.

circle arrangement

number summation

minimum value

optimization problem

mathematical constraints