MathDB

5

Part of 2014 Iran Team Selection Test

Problems(3)

question5

Source: iran tst 2014 first exam

4/14/2014
nn is a natural number. for every positive real numbers x1,x2,...,xn+1x_{1},x_{2},...,x_{n+1} such that x1x2...xn+1=1x_{1}x_{2}...x_{n+1}=1 prove that: nx1+...+nxn+1≥nx1n+...+nxn+1n\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}
inequalitiesfunctioninequalities unsolvedn-variable inequality
nice problem (Inequalities)(question 5)

Source: iran tst 2014 second exam

5/21/2014
if x,y,z>0x,y,z>0 are postive real numbers such that x2+y2+z2=x2y2+y2z2+z2x2x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2} prove that ((x−y)(y−z)(z−x))2≤2((x2−y2)2+(y2−z2)2+(z2−x2)2)((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})
inequalitiesinequalities proposed
Compact sets

Source: Iran TST 2014, third exam, day 2, problem 2

1/1/2015
Given a set X={x1,…,xn}X=\{x_1,\ldots,x_n\} of natural numbers in which for all 1<i≤n1< i \leq n we have 1≤xi−xi−1≤21\leq x_i-x_{i-1}\leq 2, call a real number aa good if there exists 1≤j≤n1\leq j \leq n such that 2∣xj−a∣≤12|x_j-a|\leq 1. Also a subset of XX is called compact if the average of its elements is a good number. Prove that at least 2n−32^{n-3} subsets of XX are compact.
Proposed by Mahyar Sefidgaran
combinatorics unsolvedcombinatorics