7th APMO 1995


A1. Find all real sequences x₁, x₂, ... , x₁₉₉₅ which satisfy 2√(x_n - n + 1) ≥ x_n+1 - n + 1 for n = 1, 2, ... , 1994, and 2√(x₁₉₉₅ - 1994) ≥ x₁ + 1.
A2. Find the smallest n such that any sequence a₁, a₂, ... , a_n whose values are relatively prime square-free integers between 2 and 1995 must contain a prime. [An integer is square-free if it is not divisible by any square except 1.]
A3. ABCD is a fixed cyclic quadrilateral with AB not parallel to CD. Find the locus of points P for which we can find circles through AB and CD touching at P.
A4. Take a fixed point P inside a fixed circle. Take a pair of perpendicular chords AC, BD through P. Take Q to be one of the four points such that AQBP, BQCP, CQDP or DQAP is a rectangle. Find the locus of all possible Q for all possible such chords.
A5. f is a function from the integers to {1, 2, 3, ... , n} such that f(A) and f(B) are unequal whenever A and B differ by 5, 7 or 12. What is the smallest possible n?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.