Question

На доске написаны четыре ненулевых числа, причём сумма любых трёх из них меньше четвёртого числа. Какое наименьшее количество отрицательных чисел может быть написано на доске?

Answer 1

Это три отрицательных числа и одно положительное.
Например, -2; -2; -2; 1.
Суммы -2-2+1=-3<-2; -2-2-2=-6<1

Answer 2

Minimum number of negative numbers that can be written on the board

To determine the minimum number of negative numbers that can be written on the board, we need to consider the given conditions: there are four non-zero numbers on the board, and the sum of any three of them is less than the fourth number.

Let's analyze the possible scenarios:

1. If all four numbers are positive, the sum of any three of them will always be greater than the fourth number. Therefore, there cannot be four positive numbers on the board.

2. If three numbers are positive and one number is negative, the sum of the three positive numbers will always be greater than the negative number. Therefore, there cannot be three positive numbers and one negative number on the board.

3. If two numbers are positive and two numbers are negative, we need to find the minimum number of negative numbers that satisfy the given conditions.

Let's assume the four numbers on the board are a, b, c, and d, where a and b are positive, and c and d are negative.

According to the given conditions, the sum of any three numbers should be less than the fourth number. Let's consider the possible combinations:

- a + b + c < d: In this case, the sum of the two positive numbers (a and b) and the negative number (c) is less than the other negative number (d). To satisfy this condition, we need at least one more negative number. Therefore, the minimum number of negative numbers that can be written on the board is 2.

- a + b + d < c: In this case, the sum of the two positive numbers (a and b) and the negative number (d) is less than the other negative number (c). To satisfy this condition, we need at least one more negative number. Therefore, the minimum number of negative numbers that can be written on the board is 2.

- a + c + d < b: In this case, the sum of the positive number (a) and the two negative numbers (c and d) is less than the other positive number (b). To satisfy this condition, we need at least one more negative number. Therefore, the minimum number of negative numbers that can be written on the board is 2.

- b + c + d < a: In this case, the sum of the positive number (b) and the two negative numbers (c and d) is less than the other positive number (a). To satisfy this condition, we need at least one more negative number. Therefore, the minimum number of negative numbers that can be written on the board is 2.

In all possible combinations, the minimum number of negative numbers that can be written on the board is 2.

Therefore, the minimum number of negative numbers that can be written on the board is 2.