MathDB

3

Part of 2011 IMC

Problems(2)

IMC 2011 Day 1 Problem 3

Source:

7/30/2011
Let pp be a prime number. Call a positive integer nn interesting if xn−1=(xp−x+1)f(x)+pg(x)x^n-1=(x^p-x+1)f(x)+pg(x) for some polynomials ff and gg with integer coefficients. a) Prove that the number pp−1p^p-1 is interesting. b) For which pp is pp−1p^p-1 the minimal interesting number?
algebrapolynomialsuperior algebrasuperior algebra unsolved
IMC 2011 Day 2, Problem 3

Source:

8/1/2011
Calculate ∑n=1∞ln⁡(1+1n)ln⁡(1+12n)ln⁡(1+12n+1)\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right).
logarithmslimitIMCcollege contests