3. The boxes have different masses. If for every two boxes we calculate the sum of their masses in kilograms, we get the following 10 numbers: 114, 85, 122, 74, 133, 118, 147, 99, 107 and 93. What is the mass of each of the five boxes?
3. The boxes have different masses. If for every two boxes we calculate the sum of their masses in kilograms, we get the following 10 numbers: 114, 85, 122, 74, 133, 118, 147, 99, 107, and 93. What is the mass of each of the five boxes?

## Solution.

Let the masses of the boxes be $a, b, c, d$, and $e$ where $a < b < c < d < e$.

When we add the masses of any two boxes, we get a total of 10 sums:

$a+b, a+c, a+d, a+e, b+c, b+d, b+e, c+d, c+e, d+e$.

Notice that each number appears in exactly 4 sums.

Therefore, the sum of all 10 sums is $4 \cdot (a+b+c+d+e)$

The sum of all ten given numbers is

$114 + 85 + 122 + 74 + 133 + 118 + 147 + 99 + 107 + 93 = 1092$

By equating the obtained sums, we have:

$4 \cdot (a+b+c+d+e) = 1092$

$a+b+c+d+e = 1092 : 4$

$a+b+c+d+e = 273$

Thus, the sum of all 5 masses is $273 \text{ kg}$.

The smallest sum is obtained by adding the two smallest numbers, which are $a$ and $b$. The largest sum is obtained by adding the two largest numbers, which are $d$ and $e$. Therefore, the two largest masses have a sum of $d+e = 147 \text{ kg}$, and the two smallest masses have a sum of $a+b = 74 \text{ kg}$.

1 POINT

By adding the equalities, we get $a+b+d+e = 221$, and since $a+b+c+d+e = 273$, it follows that $c = 273 - 221 = 52$.

1 POINT

The second largest sum is obtained by adding the smallest and the third largest, which are $a$ and $c$, so $a+c = 85$, from which it follows that $a = 85 - 52 = 33$.

$a+b = 74$ so $b = 74 - 33 = 41$

1 POINT

The second smallest sum is obtained by adding the largest and the third largest, which are $c$ and $e$, so $c+e = 133$, from which it follows that $e = 133 - 52 = 81$.

$d+e = 147$ so $d = 147 - 81 = 66$.

1 POINT

The masses of the boxes are: $33 \text{ kg}, 41 \text{ kg}, 52 \text{ kg}, 66 \text{ kg}$, and $81 \text{ kg}$.

TOTAL 10 POINTS