MathDB

4

Part of 2008 District Olympiad

Problems(6)

Cyclic quadrilater

Source: Romanian DMO 9th grade P4

3/1/2008
Let ABCD ABCD be a cyclic quadrilater. Denote P\equal{}AD\cap BC and Q\equal{}AB \cap CD. Let E E be the fourth vertex of the parallelogram ABCE ABCE and F\equal{}CE\cap PQ. Prove that D,E,F D,E,F and Q Q lie on the same circle.
geometryparallelogramtrigonometryGaussgeometric transformationreflectionratio
Set of numbers

Source: Romanian DMO 7th grade problem four

3/1/2008
Let M M be the set of those positive integers which are not divisible by 3 3. The sum of 2n 2n consecutive elements of M M is 300 300. Determine n n.
inductionnumber theory proposednumber theory
3-variable inequality

Source: Romanian DMO 8th grade P4

3/1/2008
Determine x,y,z>0 x,y,z>0 for which x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.
inequalitiesalgebra proposedalgebra
Set of functions with &quot;order&quot; 2008

Source: RMO District Round, Bucharest 2008, Grade 10, Problem 4

1/27/2008
Let A A represent the set of all functions f:N→N f : \mathbb{N} \rightarrow \mathbb{N} such that for all k∈1,2007‾ k \in \overline{1, 2007}, f[k]≠IdN f^{[k]} \neq \mathrm{Id}_{\mathbb{N}} and f[2008]≡IdN f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}. a) Prove that A A is non-empty. b) Find, with proof, whether A A is infinite. c) Prove that all the elements of A A are bijective functions. (Denote by N \mathbb{N} the set of the nonnegative integers, and by f[k] f^{[k]}, the composition of f f with itself k k times.)
functionalgebra proposedalgebra
Romania District Olympiad 2008 - Grade XI

Source:

4/10/2011
Find the values of a∈[0,∞)a\in [0,\infty) for which there exist continuous functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R}, such that f(f(x))=(x−a)2, (∀)x∈Rf(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}.
functionreal analysisreal analysis unsolved
Neighbour polynomials over finite field

Source: Romanian District Olympiad 2008, Grade XII, Problem 4

10/7/2018
Let be a finite field K. K. Say that two polynoms f,g f,g from K[X] K[X] are neighbours, if the have the same degree and they differ by exactly one coefficient.
a) Show that all the neighbours of 1+X2 1+X^2 from Z3[X] \mathbb{Z}_3[X] are reducible in Z3[X]. \mathbb{Z}_3[X] .
b) If ∣K∣≥4, |K|\ge 4, show that any polynomial of degree ∣K∣−1 |K|-1 from K[X] K[X] has a neighbour from K[X] K[X] that is reducible in K[X], K[X] , and also has a neighbour that doesn´t have any root in K. K.
algebrapolynomialsuperior algebraRing Theoryfinite fieldWedderburn