Junior Mathematical Olympiad, September 2019

Question

One hundred students wear shirts numbered from 1 to 100. The students are arranged in a square of ten rows by ten columns. It turns out that adding the ten shirt numbers of the students in any row or any column always yields the same outcome. Determine that outcome.

Accepted Answer

Let us consider the arrangement of the students in a

10 \times 10

square. The shirt numbers are

1, 2, \dots, 100

.

The sum of all shirt numbers is:

1 + 2 + \dots + 100 = \frac{100 \times 101}{2} = 5050.

There are

10

rows, and the sum of the numbers in each row is the same. Let

S

be the sum for each row. Then:

10 S = 5050 ⟹ S = 505.

Similarly, for columns, the sum is also

505

.

Answer: The outcome is

505

.