Problems

Show that given any \(3\) numbers, there will be two of them so that their difference is an even number.

The triangle \(EFG\) is isosceles with \(EF=EG\). A circle with center \(A\) is tangent to the sides \(EF\) and \(EG\) at the points \(C\) and \(B\) respectively. It is also tangent to the circle circumscribed around the triangle \(EFG\) at the point \(H\). Prove that the midpoint of the segment \(BC\) is the center of the circle inscribed into the triangle \(EFG\).

(IMO 1999) Two circles with centres \(A\) and \(C\) intersect at the points \(B\) and \(G\), moreover the circle with centre \(C\) goes through \(A\). A big circle is tangent to both given circles at the points \(E\) and \(F\) (see picture). The line \(BG\) intersects the big circle at the points \(H\) and \(I\). The segments \(EH\) and \(EI\) intersect with the circle with centre \(C\) at the points \(H\) and \(K\) respectively. Prove that the segment \(JK\) is tangent to the circle with center at \(A\).

On the first day Robinson Crusoe tied the goat with a single piece of rope by putting one peg into the ground. What shape did the goat graze?