MathDB

4

Part of 2014 Iran Team Selection Test

Problems(3)

question4

Source: iran tst 2014 first exam

4/14/2014
Find the maximum number of Permutation of set {1,2,3,...,20141,2,3,...,2014} such that for every 2 different number aa and bb in this set at last in one of the permutation bb comes exactly after aa
combinatorics unsolvedcombinatorics
The n-cubic Permutations

Source: Iran TST 2014, second exam, day 2 ,problem 1

1/1/2015
nn is a natural number. We shall call a permutation a1,…,ana_1,\dots,a_n of 1,…,n1,\dots,n a quadratic(cubic) permutation if ∀1≤i≤n−1\forall 1\leq i \leq n-1 we have aiai+1+1a_ia_{i+1}+1 is a perfect square(cube). (a)(a) Prove that for infinitely many natural numbers nn there exists a quadratic permutation. (b)(b) Prove that for no natural number nn exists a cubic permutation.
quadraticsnumber theory unsolvednumber theory
easy function equation(question 4)

Source: iran tst 2014 third exam

5/21/2014
Find all functions f:R+→R+f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+} such that x,y∈R+,x,y\in \mathbb{R}^{+}, f(yf(x+1))+f(x+1xf(y))=f(y) f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y)
functionalgebra proposedalgebrafunctional equation