MathDB

1

Part of 2001 Polish MO Finals

Problems(2)

inequality with powers

Source:

11/25/2005
Prove the following inequality: x1+2x2+3x3+...+nxn≤n(n−1)2+x1+x22+x33+...+xnnx_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n where ∀xixi>0\forall _{x_i} x_i > 0
inequalitiescalculusderivativefunctioninductioninequalities proposed
2^n a+b being always a square

Source:

1/14/2005
Assume that a,ba,b are integers and nn is a natural number. 2na+b2^na+b is a perfect square for every nn.Prove that a=0a=0.
modular arithmeticfunctionnumber theoryPerfect Square