1. Find the smallest positive integer with the following property: there does not exist
an arithmetic progression of real numbers containing exactly integers.
2. Let be a sequence of real numbers satisfying for all .
for each positive integer .
3. Let and be two circles intersecting at and . The common tangent, closer to , of and touches at and at . The tangent of at meets at , which is different from , and the extension of meets at . Prove that the circumcircle of triangle is tangent to and .
4. Determine all pairs of integers with the property that the numbers and are both perfect squares.
5. Let be a set of points in the plane such that no three are collinear and no
four concyclic. A circle will be called good if it has points of on its circumference,
points in its interior and points in its exterior. Prove that the number of
good circles has the same parity as .