1. (RUSMO/1983) Given that a pile of 100 small weights have a total weight of 500 g, and the weight of a small weight is 1g, 10 g or 50 g. Find the number of each kind of weights in the pile.
2. (ASUMO/1988) Prove that there are infinitely many positive integer solutions (x, y, z) to the equation x – y + z = 1, such that x, y, z are distinct, and any two of them have a product which is divisible by the remaining number.
3. a, b are two relatively prime positive integers. Prove that the equation ax + by = ab – a – b has no non-negative integer solution.
4. Prove that for relatively prime two positive integers a and b, the equation ax + by = c must have non-negative integer solution if c > ab ¡ a ¡ b.
5. (KIEV/1980) Multiply some natural number by 2 and then plus 1, and then carry out this operation on the resultant number, and so on. After repeating 100 times of such operations, whether the resulting number is divisible by
(i) 1980? (ii) by 1981?
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