MathDB

Partitioning the set

Source: Romanian District Olympiad 2014, Grade 5, P3

6/15/2014

Let

A=\{1,3,3^2,\ldots, 3^{2014}\}

. We obtain a partition of

A

if

A

is written as a disjoint union of nonempty subsets.
[*]Prove that there is no partition of

A

such that the product of elements in each subset is a square. [*]Prove that there exists a partition of

A

such that the sum of elements in each subset is a square.

number theory proposednumber theory

Find measure of angle

Source: Romanian District Olympiad 2014, Grade 7, P3

6/15/2014

Let

ABC

be a triangle in which

\measuredangle{A}=135^{\circ}

. The perpendicular to the line

AB

erected at

A

intersects the side

BC

at

D

, and the angle bisector of

\angle B

intersects the side

AC

at

E

. Find the measure of

\measuredangle{BED}

.

trigonometrygeometryincenterangle bisectorsimilar trianglesgeometry proposed

Congruent angles

Source: Romanian District Olympiad 2014, Grade 6, P3

6/15/2014

The points

M, N,

and

P

are chosen on the sides

BC, CA

and

AB

of the

\Delta ABC

such that

BM=BP

and

CM=CN

. The perpendicular dropped from

B

to

MP

and the perpendicular dropped from

C

to

MN

intersect at

I

. Prove that the angles

\measuredangle{IPA}

and

\measuredangle{INC}

are congruent.

geometryincentergeometry proposed

A regular hexagon

Source: Romanian District Olympiad 2014, Grade 8, P3

6/15/2014

Let

ABCDEF

be a regular hexagon with side length

a

. At point

A

, the perpendicular

AS

, with length

2a\sqrt{3}

, is erected on the hexagon's plane. The points

M, N, P, Q,

and

R

are the projections of point

A

on the lines

SB, SC, SD, SE,

and

SF

, respectively. [*]Prove that the points

M, N, P, Q, R

lie on the same plane. [*]Find the measure of the angle between the planes

(MNP)

and

(ABC)

.

geometrycircumcircle3D geometryspheretrigonometrygeometry proposed

Medians and an identity

Source: Romanian District Olympiad 2014, Grade 9, P3

6/15/2014

The medians

AD, BE

and

CF

of triangle

ABC

intersect at

G

. Let

P

be a point lying in the interior of the triangle, not belonging to any of its medians. The line through

P

parallel to

AD

intersects the side

BC

at

A_{1}

. Similarly one defines the points

B_{1}

and

C_{1}

. Prove that

\overline{A_{1}D}+\overline{B_{1}E}+\overline{C_{1}F}=\frac{3}{2}\overline{PG}

geometry proposedgeometry

A set with fixed number of elements

Source: Romanian District Olympiad 2014, Grade 10, P3

6/15/2014

Let

p

and

n

be positive integers, with

p\geq2

, and let

a

be a real number such that

1\leq a<a+n\leq p

. Prove that the set

\mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\}

has exactly

n+1

elements.

floor functionlogarithmsalgebra proposedalgebra

Matrices that commute

Source: Romanian District Olympiad 2014, Grade 11, P3

6/15/2014

[*]Let

A

be a matrix from

\mathcal{M}_{2}(\mathbb{C})

,

A\neq aI_{2}

, for any

a\in\mathbb{C}

. Prove that the matrix

X

from

\mathcal{M} _{2}(\mathbb{C})

commutes with

A

, that is,

AX=XA

, if and only if there exist two complex numbers

\alpha

and

\alpha^{\prime}

, such that

X=\alpha A+\alpha^{\prime}I_{2}

. [*]Let

A

,

B

and

C

be matrices from

\mathcal{M}_{2}(\mathbb{C})

, such that

AB\neq BA

,

AC=CA

and

BC=CB

. Prove that

C

commutes with all matrices from

\mathcal{M}_{2}(\mathbb{C})

.

linear algebramatrixcomplex numbers

Operations on a ring

Source: Romanian District Olympiad 2014, Grade 12, P3

6/15/2014

Let

(A,+,\cdot)

be an unit ring with the property: for all

x\in A

,

x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6}

[*]Let

x\in A

and let

n\geq2

be an integer such that

x^{n}=0

. Prove that

x=0

. [*]Prove that

x^{4}=x

, for all

x\in A

.

superior algebrasuperior algebra unsolved

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Problems(8)