Question 175
1) Pierre splits the integers $1,2, \ldots, 2012$ into two disjoint groups whose respective sums of elements are equal.
Without even looking at the partition chosen by Pierre, Clara then claims that one can remove two numbers from each group so that the respective sums of the remaining elements in the two groups are equal.
Prove that Clara is right.
2) Pierre splits the integers $1,2, \ldots, 20$ into two disjoint groups whose respective sums of elements are equal.
Without even looking at the partition chosen by Pierre, Clara then claims that one can remove two numbers from each group so that the respective sums of the remaining elements in the two groups are equal.
Prove that, this time, Clara might have done better to look before speaking.
Without even looking at the partition chosen by Pierre, Clara then claims that one can remove two numbers from each group so that the respective sums of the remaining elements in the two groups are equal.
Prove that Clara is right.
2) Pierre splits the integers $1,2, \ldots, 20$ into two disjoint groups whose respective sums of elements are equal.
Without even looking at the partition chosen by Pierre, Clara then claims that one can remove two numbers from each group so that the respective sums of the remaining elements in the two groups are equal.
Prove that, this time, Clara might have done better to look before speaking.