MathDB

3

Part of 2010 District Olympiad

Problems(6)

BT = 2PT wanted, 40-70-70 triangle (2010 Romania District VII P3)

Source:

5/19/2020
Consider triangle ABCABC with AB=ACAB = AC and ∠A=40o\angle A = 40 ^o. The points SS and TT are on the sides ABAB and BCBC, respectively, so that ∠BAT=∠BCS=10o\angle BAT = \angle BCS= 10 ^o. The lines ATAT and CSCS intersect at point PP. Prove that BT=2PTBT = 2PT.
anglesgeometryisoscelesequal segments
distance between parallel planes, cube related

Source: 2010 Romania District VIII P3

5/19/2020
Consider the cube ABCDA′B′C′D′ABCDA'B'C'D'. The bisectors of the angles ∠A′C′A\angle A' C'A and ∠A′AC′\angle A' AC' intersect AA′AA' and A′CA'C in the points PP, respectively SS. The point MM is the foot of the perpendicular from A′A' on CPCP , and NN is the foot of the perpendicular from A′A' to ASAS. Point OO is the center of the face ABB′A′ABB'A'
a) Prove that the planes (MNO)(MNO) and (AC′B)(AC'B) are parallel.
b) Calculate the distance between these planes, knowing that AB=1AB = 1.
geometry3D geometrycubedistanceplanes
Romania District Olympiad 2010

Source: Grade IX

3/13/2010
For any real number x x prove that: x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*
floor functionalgebra proposedalgebra
Romania District Olympiad 2010

Source: Grade X

3/13/2010
Find all the functions f:N→N f: \mathbb{N}\rightarrow \mathbb{N} such that 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n,   \forall n\in \mathbb{N}.
functioninductionalgebra proposedalgebra
Romanian District Olympiad

Source: Grade XI

3/17/2010
Let f:R→R f: \mathbb{R}\rightarrow \mathbb{R} a strictly increasing function such that f∘f f\circ f is continuos. Prove that f f is continuos.
functioninequalitiesreal analysisreal analysis unsolved
Romanian District Olympiad

Source: Grade XII

3/17/2010
Let a<c<b a < c < b be three real numbers and let f:[a,b]→R f: [a,b]\rightarrow \mathbb{R} be a continuos function in c c. If f f has primitives on each of the intervals [a,c) [a,c) and (c,b] (c,b], then prove that it has primitives on the interval [a,b] [a,b].
functioncalculusabstract algebrareal analysisreal analysis unsolved