MathDB

3

Part of 2006 Turkey MO (2nd round)

Problems(2)

Find all n for which the coefficients are all divisible by 7

Source: Turkey National Olympiad 2006 - D1 - P3

1/6/2007
Find all positive integers nn for which all coefficients of polynomial P(x)P(x) are divisible by 7,7, where P(x)=(x2+x+1)n−(x2+1)n−(x+1)n−(x2+x)n+x2n+xn+1.P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.
algebrapolynomialnumber theory unsolvednumber theory
turkey nmo 2006 q6

Source:

1/6/2007
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
geometrytrigonometrycircumcirclefunctionalgebrapolynomialRational Root Theorem