Few problems – 4


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  1.  Find all integers a,b,c satisfying 1<a<b<c such that (a-1)(b-1)(c-1) is a divisor of abc-1.
  2. Find the number of ordered pairs (x,y) of positive integers which satisfy xy=27027.
  3. The integer N consists of 2017 consecutive nines. A computer calculates N^3 = (999 . . . 999)^3. How many nines does the number N^3 contain in total ?
  4. Let ABC be an acute angled triangle such that \angle BAC=45^0. Let D be a point on AB such that CD\perp AB. Let P be an internal point of the segment CD. Prove that AP\perp BC if and only if |AP|=|BC|.
  5. Let f(x)=x^n+5x^{n-1}+3 where n>1 is an integer. Prove that f(x) cannot be expressed as the product of two non-constant polynomials with integer coefficients.
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