MathDB

1

Part of 2002 Turkey Team Selection Test

Problems(2)

ab(a+b) | a^2+ab+b^2

Source: Turkey TST 2002 - P1

4/6/2013
If ab(a+b)ab(a+b) divides a2+ab+b2a^2 + ab+ b^2 for different integers aa and bb, prove that ∣a−b∣>ab3.|a-b|>\sqrt[3]{ab}.
number theory proposednumber theory
function as sum of a linear function and a periodic function

Source: Turkey TST 2002 - P4

4/6/2013
If a function ff defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function.
[For every real number xx, if there is a real number aa such that f(a−x)+f(a+x)=2f(a)f(a-x) + f(a+x) =2f(a), the point (a,f(a))(a,f(a)) is called a center of symmetry of the function ff.]
functionsymmetryalgebra proposedalgebra