MathDB

2

Part of 1995 Vietnam Team Selection Test

Problems(2)

P(x) = x^{n + 1} + kx^n - 870x^2 + 1945x + 1995

Source: Vietnam TST 1995, Problem 2

7/27/2008
Find all integers k k such that for infinitely many integers n≥3 n \ge 3 the polynomial P(x)=xn+1+kxn−870x2+1945x+1995 P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995 can be reduced into two polynomials with integer coefficients.
algebrapolynomialmodular arithmeticsearchRational Root Theoremalgebra unsolved
Pair (n, p) of nonnegative integers is called nice

Source: Vietnam TST 1995, Problem 5

7/27/2008
For any nonnegative integer n n, let f(n) f(n) be the greatest integer such that 2^{f(n)} | n \plus{} 1. A pair (n,p) (n, p) of nonnegative integers is called nice if 2f(n)>p 2^{f(n)} > p. Find all triples (n,p,q) (n, p, q) of nonnegative integers such that the pairs (n,p) (n, p), (p,q) (p, q) and (n \plus{} p \plus{} q, n) are all nice.
modular arithmeticnumber theory unsolvednumber theory