1. It is easier to count the number of n-digit numbers (n = 1, 2, 3, 4) without 2. After counting those without the digit ‘2’, you should know that how many numbers less than or equal to 2003 has the digit ‘2’. 2. Let ABCDEF be the hexagon. You should see that the sum of distances from P to AB and from P to DE is a constant. It can be computed with the aid of trigonometry. Similar arguments work for another 2 pairs of distances.

4. It is equivalent to asking the number of integral solutions of . Since x = 0 or y = 0 yields no solution and the square function is even, we can count only the number of positive solutions and multiple it by 4 to obtain the answer. Check the parity of x and y for positive solutions.

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