MathDB

2

Part of 1997 Romania Team Selection Test

Problems(4)

Very hard ,isn't it?

Source: Romanian team selection test 1997, 1st round, problem 2

9/6/2005
Find the number of sets AA containing 99 positive integers with the following property: for any positive integer n≤500n\le 500, there exists a subset B⊂AB\subset A such that ∑b∈Bb=n\sum_{b\in B}{b}=n.
Bogdan Enescu & Dan Ismailescu
combinatorics unsolvedcombinatoricsAdditive combinatorics
All the positive integer that can write in form a^2+2b^2

Source: Romania TST 1997

10/10/2005
Suppose that AA be the set of all positive integer that can write in form a2+2b2a^2+2b^2 (where a,b∈Za,b\in\mathbb {Z} and bb is not equal to 00). Show that if pp be a prime number and p2∈Ap^2\in A then p∈Ap\in A.
Marcel Tena
modular arithmeticnumber theory solvednumber theoryDivisibilityMultiplicative NTMultiplicative order
Bijective function f: points -> lines

Source: Romanian TST 1997

4/18/2011
Let PP be the set of points in the plane and DD the set of lines in the plane. Determine whether there exists a bijective function f:P→Df: P \rightarrow D such that for any three collinear points AA, BB, CC, the lines f(A)f(A), f(B)f(B), f(C)f(C) are either parallel or concurrent.
Gefry Barad
functiongeometry unsolvedgeometry
Infinite subset of pairwise coprime integers

Source: Romanian TST 1997

9/17/2011
Let a>1a>1 be a positive integer. Show that the set of integers {a2+a−1,a3+a2−1,…,an+1+an−1,…}\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\} contains an infinite subset of pairwise coprime integers.
Mircea Becheanu
number theory proposednumber theory