${1,2,3,4,5,6,7,8,9,10}$ numbers are given and divided into 2 groups and let the groups be $A$ and $B$.Prove that $A$ or $B$ contains $a,b,a+b$.

Question

${1,2,3,4,5,6,7,8,9,10}$ numbers are given and divided into 2 groups and let the groups be $A$ and $B$.Prove that $A$ or $B$ contains $a,b,a+b$. $A$ contains none of the elements in the $B$. $a$ is not equal to $b$

Is this problem can be solved? If not ${1,2,3....n}$ what is lowest value of n?

Any help would be appreciated. Well i tried dividing into groups and i didn't found any problem but i didn't check all possible groups.