1

Part of 2007 IMC

Problems(2)

Quadratic polynomial, coefficients divisible by 5

Source: IMC 2007, Day 1, Problem 1

f

be a polynomial of degree 2 with integer coefficients. Suppose that

f(k)

is divisible by 5 for every integer

k

. Prove that all coefficients of

f

are divisible by 5.

quadraticsalgebrapolynomialmodular arithmeticIMCcollege contests

f can be rotated/translated onto cf

Source: IMC 2007, Day 2, Problem 1

f : \mathbb{R}\to \mathbb{R}

be a continuous function. Suppose that for any

c > 0

f

can be moved to the graph of

cf

using only a translation or a rotation. Does this imply that

f(x) = ax+b

for some real numbers

a

b

geometrygeometric transformationlogarithmsfunctionrotationIMCcollege contests