MathDB

4

Part of 2013 IMC

Problems(2)

IMC 2013/1/4

Source: Imc 2013 Problem 4

8/8/2013
Let n⩾3\displaystyle{n \geqslant 3} and let x1,x2,...,xn\displaystyle{{x_1},{x_2},...,{x_n}} be nonnegative real numbers. Define A=∑i=1nxi,B=∑i=1nxi2,C=∑i=1nxi3\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }. Prove that: (n+1)A2B+(n−2)B2⩾A4+(2n−2)AC.\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.
Proposed by Géza Kós, Eötvös University, Budapest.
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IMC 2013/2/4

Source: Imc 2013 Problem 9

8/9/2013
Does there exist an infinite set M\displaystyle{M} consisting of positive integers such that for any a,b∈M\displaystyle{a,b \in M} with a<b\displaystyle{a < b} the sum a+b\displaystyle{a + b} is square-free? Note. A positive integer is called square-free if no perfect square greater than 1\displaystyle{1} divides it.
Proposed by Fedor Petrov, St. Petersburg State University.
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