Zero joint subsums of integers

I found thousands of counter examples by using a java code which I'm gonna give at the end (I hope that the code is true because it was just brute force) one counter-ex:

[2, 2, -1, -2, 1, -2, -2, 2, -2, 2]

[2, 2, -2, -1, -2, -1, -1, 2, -1, 2]

which can be seen easily (I chose this one as easiest) Because two sequence have 4 common 2's, and for each -1 of second sequence there is -2 in the first sequence with corresponding indices

So, I think you see what I see :)

some other counter examples:

[-2, 2, 0, -1, 0, 2, -2, 0, -1, 2]

[2, -1, 1, -2, 1, -1, 2, 1, -2, -1]

//////////////

[2, -2, -2, 1, -2, -2, 2, -1, 2, 2]

[-2, 1, 1, -2, 1, 1, 2, -2, 2, -2]

And here is the code with a lot brute force:

It is a bit messy but you can arrange the indentation if you want to run the code

import java.util.; import java.io.;

public class numbers {

public static void main(String[] args) {
    ArrayList<int[]> myList = new ArrayList<int[]>() ;
    Random gen= new Random();
    while (myList.size()<90045){
        int[] arr = new int[10];
        boolean itsdone = false ;
        for (int i=0; i<10 ; i++){
            arr[i]=gen.nextInt(5)-2;
        }
        while (myList.contains(arr) || (summ(arr)!=0) ){
            for (int i=0; i<10 ; i++){
                arr[i]=gen.nextInt(5)-2;
            }
        }
        if (!myList.contains(arr) && summ(arr)==0 )
            myList.add(arr);
            System.out.println(myList.size());
    }
    for (int i=0 ; i<myList.size();i++){
        for (int j=i; j<myList.size();j++)
            compareArr(myList.get(i),myList.get(j));
    }    

} `public static int summ(int[] arr) {

    int summ=0;
    for (int i=0; i<arr.length ; i++)
        summ+=arr[i];   
    return summ;`


}

public static boolean compareArr(int[] arr1,int[] arr2){ for (int i=0; i<10 ; i++){ if (arr1[i]==0 && arr2[i]==0) return true; for (int j=i+1; j<10 ; j++){ if ((arr1[i]+arr1[j]==0) && (arr2[i]+arr2[j])==0) return true; for (int k=j+1; k<10 ; k++){ if ((arr1[i]+arr1[j]+arr1[k]==0) && (arr2[i]+arr2[j]+arr2[k])==0) return true; for (int t=k+1; t<10 ; t++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t])==0) return true; for (int m=t+1; m<10 ; m++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]+arr1[m]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t]+arr2[m])==0) return true; for (int n=m+1; n<10 ; n++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]+arr1[m]+arr1[n]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t]+arr2[m]+arr2[n])==0) return true; for (int a=n+1; a<10 ; a++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]+arr1[m]+arr1[n]+arr1[a]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t]+arr2[m]+arr2[n]+arr2[a])==0) return true; for (int b=a+1; b<10 ; b++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]+arr1[m]+arr1[n]+arr1[a]+arr1[b]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t]+arr2[m]+arr2[n]+arr2[a]+arr2[b])==0) return true; for (int c=b+1; c<10 ; c++){ if ((arr1[i]+arr1[j]+arr1[k]+arr1[t]+arr1[m]+arr1[n]+arr1[a]+arr1[b]+arr1[c]==0) && (arr2[i]+arr2[j]+arr2[k]+arr2[t]+arr2[m]+arr2[n]+arr2[a]+arr2[b]+arr2[c])==0) return true;

                }   */
                }
                }
                }
                }
                }   
                }
                }
            }
        }
        System.out.println(Arrays.toString(arr1));
        System.out.println(Arrays.toString(arr2));
        System.out.println("--------------");
        return false;

   } 
}

(Added by PL) Countexamples for $x_1,y_1,\ldots,x_{11},y_{11}$ (see comments below):

[0, -2, -2, -2, -2, 1, 2, 1, 1, 2, 1] [-2, -1, -1, -1, -1, 2, -1, 2, 2, -1, 2]

[0, 1, -2, -2, 0, 2, -2, -2, 1, 2, 2] [1, -2, -1, -1, 1, 2, -1, -1, -2, 2, 2]

[1, -1, -2, -1, -2, 2, -1, 2, 1, -1, 2] [0, 2, -1, 2, -1, -2, 2, -2, 0, 2, -2]

Countexamples for $x_1,y_1,\ldots,x_{12},y_{12}$:

[2, 2, 2, 2, -1, -2, -1, 2, -2, -2, 0, -2] [-1, -1, -1, -1, -2, 2, -2, -1, 2, 2, 1, 2] [-2, -1, 2, -2, -1, 2, -1, 2, -1, -1, 1, 2] [1, -2, 2, 1, -2, 2, -2, 2, -2, -2, 0, 2]

Countexamples for $x_1,y_1,\ldots,x_{13},y_{13}$:

[-2, -2, 2, 2, -1, 2, -1, -1, -1, 2, -1, 2, -1] [1, 1, 2, 2, -2, 2, -2, -2, -2, 2, -2, 2, -2]