MathDB

4

Part of 2006 IMC

Problems(2)

IMC 2006, problem 4, day 1

Source:

7/22/2006
Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that f(n)f(n) is an integer for infinitely many integers n. Prove that f is a polynomial.
functionalgebrapolynomialrational functionIMCcollege contests
IMC 2006 / B4

Source: IMC 2006 day 2 problem 4

7/26/2006
Let v0v_{0} be the zero ector and let v1,...,vn+1∈Rnv_{1},...,v_{n+1}\in\mathbb{R}^{n} such that the Euclidian norm ∣vi−vj∣|v_{i}-v_{j}| is rational for all 0≤i,j≤n+10\le i,j\le n+1. Prove that v1,...,vn+1v_{1},...,v_{n+1} are linearly dependent over the rationals.
linear algebramatrixvectorIMCcollege contests