1

Problems(2)

Simple Inequality

Source: Pre-VMO 2012 - Round 2 - Problem 1

12/25/2011

For

a,b,c>0: \; abc=1

prove that

a^3+b^3+c^3+6 \ge (a+b+c)^2

inequalitieslogarithmscalculusderivativeinequalities proposed

2^n-1 | Sigma(2^n_i) => k>=n

Source: Pre-VMO 2012 - Round 2 - Problem 5

12/26/2011

Let

n \geq 2

be a positive integer. Suppose there exist non-negative integers

{n_1},{n_2},\ldots,{n_k}

such that

2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}

. Prove that

k \ge n

modular arithmeticnumber theory proposednumber theory