Exercise 1. Martin wrote the pair of integers $(1011,1012)$ on the board. Then, every minute, if the pair $(a, b)$ is written on the board, he erases it and replaces it, according to his choice, with one of the pairs $(b, a)$, $(b+1, a-1)$, or $(b-2, a+2)$, with the only constraint being to write only pairs where both numbers are non-negative.

What pairs of integers can Martin write on the board after a finite number of such operations?
Solution to Exercise 1: No operation changes the sum of the integers written on the board, so Martin can only write pairs of the form $(a, 2013-a)$, with $0 \leqslant a \leqslant 2013$.

Furthermore, if Martin starts from a pair $(a, b)$ for which $a \geqslant 1$, he can successively write the pairs $(b+1, a-1)$ and $(a-1, b+1)$. Thus, by induction on $k$, Martin can write any pair of the form $(1011-k, 1012+k)$ when $0 \leqslant k \leqslant 1011$.

Similarly, if Martin starts from a pair $(a, b)$ for which $b \geqslant 1$, he can successively write the pairs $(b, a)$ and then $(a+1, b-1)$. Thus, by induction on $k$, Martin can write any pair of the form $(1011+k, 1012-k)$ when $0 \leqslant k \leqslant 1012$.

In conclusion, the pairs that Martin can write are precisely those of the form $(a, 2013-a)$, with $0 \leqslant a \leqslant 2013$.

Graders' Comments: The exercise was generally well done. However, many students stopped halfway through their solution.

When we ask for the set of pairs that Martin can write, we need to do three things:

1. provide a reasonably simple description of the pairs in question (if a student wrote "these are the pairs he can write," the student would technically be correct, but would obviously get no points);
2. prove that Martin can indeed manage to write each of these pairs;
3. prove that Martin will never be able to write a pair other than those described.

In practice, many students forgot step 1 and simply listed pairs that Martin could write, possibly with ellipses, but without making it clear how to interpret these ellipses and the pairs they were supposed to represent.

Moreover, a substantial number of students forgot step 2, or simply stated that "Martin could obviously write all the relevant pairs," which does not constitute a sufficient justification since it is one of the two main steps of the exercise.

Finally, many students forgot step 3 and effectively described how Martin could obtain each of the pairs $(a, 2023-a)$, but without even considering whether he could write other pairs; often, these students focused only on Martin's first two operations, but not the third, which involved transforming a pair $(a, b)$ into $(b-2, a+2)$.

It is crucial, in a similar situation, not to neglect any of steps 1 to 3.