## 30 Aug Few Problems – 1

- Given a straight line and points and on the same side of it find the shortest path from to which touches .
- Given a function such that

Show that for all . Thus, infer that for such a function knowing its value at one non-zero point is sufficient for finding its values at any point. - Find all integer values for , and satisfying the equation

. - Given a triangle , let , and be its altitudes. Show that , and are concurrent i.e. they meet a single point.
- For a given prime number find the number of quadruples such that (mod ).
- 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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