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1. Fix a positive integer \(\displaystyle n\) and write the numbers \(\displaystyle 0, 1, \ldots, n-1\) on a whiteboard in some order. Two numbers form an inversion if the greater precedes the smaller. A number \(\displaystyle k\) is called special if \(\displaystyle k\) forms inversions with exactly \(\displaystyle k\) other numbers. At most how many special numbers can we have on the board?
2. Numbers in this problem are written in base \(\displaystyle 10\). We allow numbers starting with zero. We call a number with an even number of digits cuttable if after cutting the number into two numbers of the same length, the square of the sum of the parts is the original number. For example, \(\displaystyle 2025=(20+25)^2\) and \(\displaystyle 0001=(00+01)^2\) are \(\displaystyle 4\)-digit cuttable numbers. Prove that the number of \(\displaystyle 2n\)-digit cuttable numbers is a power of \(\displaystyle 2\) for any \(\displaystyle n\).
3. Assume that \(\displaystyle n\ge 10\) points are given on the plane, no three being collinear. Prove that it is possible to color them red and blue so that any halfplane containing at least \(\displaystyle 10\) points contains a red and a blue point.
