APMO 1998

Problems

**Problem 1:** Let *F* be the set of all *n*-tuples (*A*_{1}, *A*_{2},…,*A _{n}*) where each

.

**Problem 2:** Show that for any positive integers a and b, (36*a* + *b*)(*a* + 36*b*) cannot be a power of 2.

**Problem 3:** Let *a*, *b*, *c* be positive real numbers. Prove that

**Problem 4:** Let *ABC* be a triangle and *D* the foot of the altitude from *A*. Let *E* and *F* be on a line passing though *D* such that *AE* is perpendicular to *BE*, *AF* is perpendicular to *CF*, and *E* and *F* are different from *D*. Let *M* and *N* be the midpoints of the line segments *BC* and *EF*, respectively. Prove that *AN* is perpendicular to *NM*.

**Problem 5:** Determine the largest of all integers n with the property that n is divisible by all positive integers that are less than .