### 7th Putnam 1947

**Problem A3**

ABC is a triangle and P an interior point. Show that we cannot find a piecewise linear path K = K_{1}K_{2} ... K_{n} (where each K_{i}K_{i+1} is a straight line segment) such that: (1) none of the K_{i} do not lie on any of the lines AB, BC, CA, AP, BP, CP; (2) none of the points A, B, C, P lie on K; (3) K crosses each of AB, BC, CA, AP, BP, CP just once; (4) K does not cross itself.

**Solution**

Each time K crosses the boundary of a triangle it moves from the outside to the inside or vice versa. K has two endpoints, so we can find one of the three triangles ABP, BCP, CAP in which it does not start or finish. But that is impossible - on the first crossing it must go from outside to inside, on the second from inside to outside and on the third from outide to inside.

7th Putnam 1947

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002