Few Problems - 3

Few Problems
  1. Let ABC be an acute angled triangle such that \angle BAC = 45 ^o. 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|.
  2. Show that there are infinitely many positive integers n such that n|3^{n}-1.
  3. Show that n^5+n^4+1 is always composite for all natural numbers n > 1. (Hint : Factorise x^5+x^4+1)
  4. Show that the number of natural numbers divisible by k\in \mathbb{N} less than or equal to n\in\mathbb{N} is \lfloor n/k \rfloor.
  5. For real numbers a,b,c>0  prove that
    3(a^2+b^2+c^2)\geq (a+b+c)^2\geq 3(ab+bc+ca).


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