MathDB

1

Part of 2006 Romania National Olympiad

Problems(6)

MN and NP are perpendicular if and only if PN is bisector

Source: Romanian NMO 2006, Grade 7, Problem 1

4/18/2006
Let ABCABC be a triangle and the points MM and NN on the sides ABAB respectively BCBC, such that 2⋅CNBC=AMAB2 \cdot \frac{CN}{BC} = \frac{AM}{AB}. Let PP be a point on the line ACAC. Prove that the lines MNMN and NPNP are perpendicular if and only if PNPN is the interior angle bisector of ∠MPC\angle MPC.
geometryangle bisector
Prism with six faces circumscriptible quadrilaterals

Source: Romanian NMO 2006, Grade 8, Problem 1

4/18/2006
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
geometry3D geometryprism
(x^3+1)(y^3+1) results in ugly ineq

Source: RMO 2006

4/17/2006
Find the maximal value of (x3+1)(y3+1), \left( x^3+1 \right) \left( y^3 + 1\right) , where x,y∈Rx,y \in \mathbb R, x+y=1x+y=1. Dan Schwarz
calculusderivativefunctioninequalities proposedinequalities
Function on the power set

Source: RMO 2006, 10th grade

4/17/2006
Let M\displaystyle M be a set composed of n\displaystyle n elements and let P(M)\displaystyle \mathcal P (M) be its power set. Find all functions f:P(M)→{0,1,2,…,n}\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \} that have the properties (a) f(A)≠0\displaystyle f(A) \neq 0, for A≠ϕ\displaystyle A \neq \phi; (b) f(A∪B)=f(A∩B)+f(AΔB)\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right), for all A,B∈P(M)\displaystyle A,B \in \mathcal P (M), where AΔB=(A∪B)\(A∩B)\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right).
functioninductionalgebra proposedalgebra
Equivalent condition for 1+1=0

Source: RMO 2006, 12th grade

4/17/2006
Let K\displaystyle \mathcal K be a finite field. Prove that the following statements are equivalent: (a) 1+1=0\displaystyle 1+1=0; (b) for all f∈K[X]\displaystyle f \in \mathcal K \left[ X \right] with deg f≥1\displaystyle \textrm{deg} \, f \geq 1, f(X2)\displaystyle f \left( X^2 \right) is reducible.
abstract algebraalgebrapolynomialgroup theorysuperior algebrasuperior algebra solved
The adjoint

Source: RMO 2006

4/17/2006
Let AA be a n×nn\times n matrix with complex elements and let A⋆A^\star be the classical adjoint of AA. Prove that if there exists a positive integer mm such that (A⋆)m=0n(A^\star)^m = 0_n then (A⋆)2=0n(A^\star)^2 = 0_n. Marian Ionescu, Pitesti
linear algebramatrixvectorlinear algebra unsolved