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?

I take a fancy to problem model this. Thank you.

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