Few problems - 4

  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 data-recalc-dims=1" /> is an integer. Prove that f(x) cannot be expressed as the product of two non-constant polynomials with integer coefficients.
Print Friendly, PDF & Email
The following two tabs change content below.

Debashish Sharma

Assistant Professor at Gurucharan College
Dr. Debashish Sharma is an Assistant Professor in the Department of Mathematics, Gurucharan College, Silchar, Assam.

Latest posts by Debashish Sharma (see all)

READ:   Book Review: My Search for Ramanujan
1Comment

Post A Comment