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Problem 515

AMC 10/12, early questions

Algebra Difficulty 3.1 Prove it Junior Mathematical Olympiad, September · Netherlands · 2019

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.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let us consider the arrangement of the students in a $10 \times 10$ square. The shirt numbers are $1, 2, \ldots, 100$ .

The sum of all shirt numbers is:
$1 + 2 + \cdots + 100 = \frac{100 \times 101}{2} = 5050.$

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:
$10S = 5050 \implies S = 505.$

Similarly, for columns, the sum is also $505$ .

Answer:
The outcome is $505$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.