MathDB

Easy circle geometry from Romania

Source: Romanian NMO 2021 grade 7 P1

4/15/2023

Let

\mathcal C

be a circle centered at

O

and

A\ne O

be a point in its interior. The perpendicular bisector of the segment

OA

meets

\mathcal C

at the points

B

and

C

, and the lines

AB

and

AC

meet

\mathcal C

again at

D

and

E

, respectively. Show that the circles

(OBC)

and

(ADE)

have the same centre.
Ion Pătrașcu, Ion Cotoi

geometryperpendicular bisectorcircle

Classical solid geo from Romania

Source: Romanian NMO 2021 grade 8 P1

4/15/2023

In the cuboid

ABCDA'B'C'D'

with

AB=a

,

AD=b

and

AA'=c

such that

a>b>c>0

, the points

E

and

F

are the orthogonal projections of

A

on the lines

A'D

and

A'B

, respectively, and the points

M

and

N

are the orthogonal projections of

C

on the lines

C'D

and

C'B

, respectively. Let

DF\cap BE=\{G\}

and

DN\cap BM=\{P\}

.
[*] Show that

(A'AG)\parallel (C'CP)

and determine the distance between these two planes; [*] Show that

GP\parallel (ABC)

and determine the distance between the line

GP

and the plane

(ABC)

.
Petre Simion, Nicolae Victor Ioan

solid geometrygeometry

sin 2B = cot C

Source: Romania NMO 2021 grade 9 P1

4/15/2023

Let

ABC

be an acute-angled triangle with the circumcenter

O

. Let

D

be the foot of the altitude from

A

. If

OD\parallel AB

, show that

\sin 2B = \cot C

.
Mădălin Mitrofan

geometrycircumcircle

three interesting complex numbers

Source:

4/25/2021

Find the complex numbers

x,y,z

,with

\mid x\mid=\mid y\mid=\mid z\mid

,knowing that

x+y+z

and

x^{3}+y^{3}+z^{3}

are be real numbers.

complex numbersalgebra

Romania National Olympiad Grade 11 P1

Source:

4/28/2021

Let

f:[a,b] \rightarrow \mathbb{R}

a function with Intermediate Value property such that

f(a) * f(b) < 0

. Show that there exist

\alpha

,

\beta

such that

a < \alpha < \beta < b

and

f(\alpha) + f(\beta) = f(\alpha) * f(\beta)

.

real analysis

Romanian NMO 2021

Source:

4/25/2021

Find all continuous functions

f:\left[0,1\right]\rightarrow[0,\infty)

such that:

\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}

integralscollege contests

1

Problems(6)