MathDB

2

Part of 2006 District Olympiad

Problems(6)

A triangle ABC with ABC = 2<ACB

Source: Romanian District Olympiad 2006, Grade 7, Problem 2

3/11/2006
In triangle ABCABC we have ∠ABC=2∠ACB\angle ABC = 2 \angle ACB. Prove that a) AC2=AB2+AB⋅BCAC^2 = AB^2 + AB \cdot BC; b) AB+BC<2⋅ACAB+BC < 2 \cdot AC.
inequalitiestrigonometry
Sum of fractions

Source: Romanian District Olympiad 2006, Grade 8, Problem 2

3/11/2006
For a positive integer nn we denote by u(n)u(n) the largest prime number less than or equal to nn, and with v(n)v(n) the smallest prime number larger than nn. Prove that 1u(2)v(2)+1u(3)v(3)+⋯+1u(2010)v(2010)=12−12011. \frac 1 {u(2)v(2)} + \frac 1{u(3)v(3)} + \cdots + \frac 1{ u(2010)v(2010)} = \frac 12 - \frac 1{2011}.
9x9 square filled with positive integers from 1 to 81

Source: Romanian District Olympiad 2006, Grade 9, Problem 2

3/11/2006
A 9×99\times 9 array is filled with integers from 1 to 81. Prove that there exists k∈{1,2,3,…,9}k\in\{1,2,3,\ldots, 9\} such that the product of the elements in the row kk is different from the product of the elements in the column kk of the array.
combinatorics proposedcombinatorics
ABC equilateral if MNP equilateral

Source: Romanian District Olympiad 2006, Grade 10, Problem 2

3/11/2006
Let ABCABC be a triangle and let M,N,PM,N,P be points on the sides BCBC, CACA and ABAB respectively such that APPB=BMMC=CNAN. \frac{AP}{PB} = \frac{BM}{MC} = \frac{CN}{AN}. Prove that triangle if MNPMNP is equilateral then triangle ABCABC is equilateral.
trigonometry
I-a^p

Source: Romanian MO 2006, District Round

3/11/2006
Let n,p≥2n,p \geq 2 be two integers and AA an n×nn\times n matrix with real elements such that Ap+1=AA^{p+1} = A. a) Prove that rank(A)+rank(In−Ap)=n\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n. b) Prove that if pp is prime then rank(In−A)=rank(In−A2)=…=rank(In−Ap−1). \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) .
linear algebramatrixinequalitiesalgebrapolynomialabstract algebralinear algebra unsolved
Groups of matrices not isomorphical

Source: Romanian District Olympiad 2006, Grade 12, Problem 2

3/11/2006
Let G={A∈M2(C)∣∣det⁡A∣=1}G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \} and H={A∈M2(C)∣det⁡A=1}H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}. Prove that GG and HH together with the operation of matrix multiplication are two non-isomorphical groups.
linear algebramatrixgroup theoryabstract algebrainvariantsuperior algebrasuperior algebra solved