MathDB

3

Part of 2013 Romania Team Selection Test

Problems(5)

Injective function with sum of reciprocal integer

Source: Romania TST 1 P3, 2013

4/5/2013
Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If SS is a finite set of positive integers such that ∑s∈S1s\sum\limits_{s\in S}\frac{1}{s} is an integer, then ∑s∈S1f(s)\sum\limits_{s\in S}\frac{1}{f\left( s\right) } is also an integer.
functioninductioninequalitiesalgebra unsolvedalgebra
Infinite number of primes expressible in a given form

Source: Romania TST 2013 Test 2 Problem 3

4/26/2013
Let SS be the set of all rational numbers expressible in the form (a12+a1−1)(a22+a2−1)…(an2+an−1)(b12+b1−1)(b22+b2−1)…(bn2+bn−1)\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)} for some positive integers n,a1,a2,…,an,b1,b2,…,bnn, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n. Prove that there is an infinite number of primes in SS.
quadraticsinequalitiesinductionnumber theory proposednumber theory
Find the maximal !

Source: Romania TST 2013 Day 3 Problem 3

1/21/2015
Determine the largest natural number rr with the property that among any five subsets with 500500 elements of the set {1,2,…,1000}\{1,2,\ldots,1000\} there exist two of them which share at least rr elements.
combinatorics unsolvedcombinatorics
Polynomial equation!

Source: Romania TST 2013 Day 4 Problem 3

1/21/2015
Given an integer n≥2n\geq 2, determine all non-constant polynomials ff with complex coefficients satisfying the condition 1+f(Xn+1)=f(X)n.1+f(X^n+1)=f(X)^n.
algebrapolynomialalgebra unsolved
Triangular array!

Source: Romania TST 2013 Day 5 Problem 3

1/21/2015
Given a positive integer nn, consider a triangular array with entries aija_{ij} where ii ranges from 11 to nn and jj ranges from 11 to n−i+1n-i+1. The entries of the array are all either 00 or 11, and, for all i>1i > 1 and any associated jj , aija_{ij} is 00 if ai−1,j=ai−1,j+1a_{i-1,j} = a_{i-1,j+1}, and aija_{ij} is 11 otherwise. Let SS denote the set of binary sequences of length nn, and define a map f ⁣:S→Sf \colon S \to S via f ⁣:(a11,a12,⋯ ,a1n)→(an1,an−1,2,⋯ ,a1n)f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n}). Determine the number of fixed points of ff.
combinatorics unsolvedcombinatorics