MathDB

3

Part of 2006 District Olympiad

Problems(6)

A connected set of 4 positive integers

Source: Romanian District Olympiad 2006, Grade 7, Problem 3

3/11/2006
A set MM of positive integers is called connected if for any element x∈Mx\in M at least one of the numbers x−1,x+1x-1,x+1 is in MM. Let UnU_n be the number of the connected subsets of {1,2,…,n}\{1,2,\ldots,n\}. a) Compute U7U_7; b) Find the smallest number nn such that Un≥2006U_n \geq 2006.
floor functionfunction
Infinity of irrationals such that x+y=xy is an integer

Source: Romanian District Olympiad 2006, Grade 8, Problem 3

3/11/2006
Prove that there exists an infinity of irrational numbers x,yx,y such that the number x+y=xyx+y=xy is a nonnegative integer.
algebrapolynomial
Binary prism

Source: Romanian District Olympiad 2006, Grade 10, Problem 3

3/11/2006
We say that a prism is binary if there exists a labelling of the vertices of the prism with integers from the set {−1,1}\{-1,1\} such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to −1-1. a) Prove that any binary prism has the number of total vertices divisible by 8; b) Prove that any prism with 2000 vertices is binary.
geometry3D geometryprismcombinatorics proposedcombinatorics
ABCD convex, BE=BD/3, AF=AC/3 => parallelogram

Source: Romanian District Olympiad 2006, Grade 9, Problem 3

3/11/2006
Let ABCDABCD be a convex quadrilateral, MM the midpoint of ABAB, NN the midpoint of BCBC, EE the intersection of the segments ANAN and BDBD, FF the intersection of the segments DMDM and ACAC. Prove that if BE=13BDBE = \frac 13 BD and AF=13ACAF = \frac 13 AC, then ABCDABCD is a parallelogram.
geometryparallelogramgeometry proposed
Bounded recurrent sequence of real numbers

Source: Romanian District Olympiad 2006, Grade 11, Problem 3

3/11/2006
Let {xn}n≥0\{x_n\}_{n\geq 0} be a sequence of real numbers which satisfy (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0,   n\geq 0. a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?
probabilityreal analysisreal analysis unsolved
Polynomial of degree n in A[X] without roots in A

Source: Romanian District Olympiad 2006, Grade 12, Problem 3

3/11/2006
Prove that if AA is a commutative finite ring with at least two elements and nn is a positive integer, then there exists a polynomial of degree nn with coefficients in AA which does not have any roots in AA.
algebrapolynomialsuperior algebrasuperior algebra solved