MathDB

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Part of 2011 Vietnam National Olympiad

Problems(2)

Prove the inequality for positive real x - [VMO 2011]

Source:

1/11/2011
Prove that if x>0x>0 and n∈N,n\in\mathbb N, then we have xn(xn+1+1)xn+1≤(x+12)2n+1.\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.
inequalitiesinequalities proposed
Sequence; prove that 2011|[a_{2012}-2010]-[VMO 2011]

Source:

1/12/2011
Define the sequence of integers ⟨an⟩\langle a_n\rangle as; a_0=1,   a_1=-1,   \text{ and }   a_n=6a_{n-1}+5a_{n-2}   \forall n\geq 2. Prove that a2012−2010a_{2012}-2010 is divisible by 2011.2011.
modular arithmeticquadraticsalgebrapolynomialnumber theorynumber theory proposed