Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.
Problem
Source: Dutch IMO TST 2015 day 2 p4
Tags: maximum value, maximum, combinatorics, Sum, Coloring
30.08.2019 22:07
31.08.2019 18:18
I know a little about combinatorics , so it might sound a little silly to those who know a lot, common sense says that the biggest sum of different $2$ numbers from $1$ to $2014$ is $2013+2014=4027$, and can be obtained in the obvious case. How come $4023$ is the solution, since $4023<4027$? How can $4023$ be the maximum, if there is at least one case where I can find a bigger value than it? PS. the official solution, also gives $4023$ as the answer.
31.08.2019 19:11
parmenides51 wrote: I know a little about combinatorics , so it might sound a little silly to those who know a lot, common sense says that the biggest sum of different $2$ numbers from $1$ to $2014$ is $2013+2014=4027$, and can be obtained in the obvious case. How come $4023$ is the solution, since $4023<4027$? How can $4023$ be the maximum, if there is at least one case where I can find a bigger value than it? PS. the official solution, also gives $4023$ as the answer. I think this problem is to ask the max value of number $\#\{k| k=m+n, m\text{ is red}, n\text{ is blue}\}$
31.08.2019 22:34
thanks, I think I understand now what it means, the lack of the red s below, probably a typo of their translation, were it there wouldn't have confused me it asks for the largest possible number of the sums (how many different sums we can have) and not the largest sum I believe this below should be the correct wording, with the red s at the end of the word integer, what do you think? Quote: Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.
01.09.2019 19:05
parmenides51 wrote: thanks, I think I understand now what it means, the lack of the red s below, probably a typo of their translation, were it there wouldn't have confused me it asks for the largest possible number of the sums (how many different sums we can have) and not the largest sum I believe this below should be the correct wording, with the red s at the end of the word integer, what do you think? Quote: Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained. Yes, the problem asks us to find the greatest number of different numbers expressible as a sum of a red and a blue number. So the corrected one is the original.
07.03.2021 12:39
Hi, Where can I find this Dutch IMO TST 2015 official solution? I`ve search all over the place but no succes. Thank you
07.03.2021 17:45
page 33 here