Few Problems

Few Problems - 1

  1. Given a straight line l and points A and B on the same side of it find the shortest path from A to B which touches l.
  2. Given a function f:\mathbb{Q}\rightarrow\mathbb{Q} such that
    f(x+y)=f(x)+f(y)

    Show that f(x)=xf(1) for all x\in\mathbb{Q}. Thus, infer that for such a function knowing its value at one non-zero point is sufficient for finding its values at any point.
  3. Find all integer values for x, y and z satisfying the equation
    x^2+y^2+z^2=2xyz
  4. Given a triangle ABC, let AP, BQ and CR be its altitudes. Show that AP, BQ and CR are concurrent i.e. they meet a single point.
  5. For a given prime number p find the number of quadruples (a,b,c,d) such that ad-bc\neq 0(mod p).
  6. There are six cities. Any two of them are connected by either a road or a railway track but not both. Show that there are at least three cities among these which are connected by a common mode of transport.

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READ:   Assam State Mathematics Olympiad question paper 2014 (Category-I)
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