Solutions to the 75 th William Lowell Putnam Mathematical Competition
نویسنده
چکیده
If n− 1 is prime, then the lowest-terms numerator is clearly either 1 or the prime n−1 (and in fact the latter, since n−1 is relatively prime to n and to (n−2)!). If n− 1 is composite, either it can be written as ab for some a 6= b, in which case both a and b appear separately in (n− 2)! and so the numerator is 1, or n− 1 = p2 for some prime p, in which case p appears in (n− 2)! and so the numerator is either 1 or p. (In the latter case, the numerator is actually 1 unless p = 2, as in all other cases both p and 2p appear in (n−2)!.)
منابع مشابه
Solutions to the 64 th William Lowell Putnam Mathematical Competition Saturday , December 6 , 2003
متن کامل
Solutions to the 66 th William Lowell Putnam Mathematical Competition
We now construct the bijection. Given a rook tour P from (1, 1) to (n, 1), let S = S(P ) denote the set of all i ∈ {1, 2, . . . , n} for which there is either a directed edge from (i, 1) to (i, 2) or from (i, 3) to (i, 2). It is clear that this set S includes n and must contain an even number of elements. Conversely, given a subset S = {a1, a2, . . . , a2r = n} ⊂ {1, 2, . . . , n} of this type ...
متن کاملThe 69 th William Lowell Putnam Mathematical Competition
A3 Start with a finite sequence a1, a2, . . . , an of positive integers. If possible, choose two indices j < k such that aj does not divide ak, and replace aj and ak by gcd(aj , ak) and lcm(aj , ak), respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: gcd means greatest common divisor and lcm means leas...
متن کاملSolutions to the 69 th William Lowell Putnam Mathematical Competition
A–1 The function g(x) = f (x,0) works. Substituting (x,y,z) = (0,0,0) into the given functional equation yields f (0,0) = 0, whence substituting (x,y,z) = (x,0,0) yields f (x,0) + f (0,x) = 0. Finally, substituting (x,y,z) = (x,y,0) yields f (x,y) = − f (y,0) − f (0,x) = g(x)−g(y). Remark: A similar argument shows that the possible functions g are precisely those of the form f (x,0) + c for som...
متن کامل