MathDB

comp geo with rectangle and 15^o - 1998 Romania NMO VII p4

Source:

8/14/2024

Let

ABCD

be a rectangle and let

E \in (BD)

such that

m( \angle DAE) =15^o

. Let

F \in AB

such that

EF \perp AB

. It is known that

EF=\frac12 AB

and

AD = a

. Find the measure of the angle

\angle EAC

and the length of the segment

(EC)

.

geometryrectangle

centroids of triangles, tetrahedron - 1998 Romania NMO VII p4

Source:

8/14/2024

Let

ABCD

be an arbitrary tetrahedron. The bisectors of the angles

\angle BDC

,

\angle CDA

and

\angle ADB

intersect

BC

,

CA

and

AB

, in the points

M

,

N

,

P

, respectively.
a) Show that the planes

(ADM)

,

(BDN)

and

(CDP)

have a common line

d

.
b) Let the points

A' \in (AD)

,

B' \in (BD)

and

C' \in (CD)

be such that

(AA') = (BB') = (CC')

; show that if

G

and

G'

are the centroids of

ABC

and

A'B'C'

then the lines

GG'

and

d

are either parallel or identical.

geometry3D geometrytetrahedron

sum sin^2 (<A_iMA_{i+1}) / d(M,A_iA_{i+1})= ... in tetrahedron

Source: 1998 Romania NMO IX p4

8/14/2024

Let

A_1A_2...A_n

be a regular polygon (

n > 4

),

T

be the common point of

A_1A_2

and

A_{n-1}A_n

and

M

be a point in the interior of the triangle

A_1A_nT

. Show that the equality

\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n}

holds if and only if

M

belongs to the circumcircle of the polygon.

geometrytetrahedrontrigonometry3D geometry

Very beautiful and easy

Source: Romanian mo 1998

12/10/2005

Suppse that

n\geq 2

and

0<x_1<x_2<...<x_n

are integer numbers. We denote that :

S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n.

(where

A

is a non-empty subset). Show that if

S_n ,S_{n-1}

were positive integer numbers , then

\forall k : S_k

is a positive integer.

algebra proposedalgebra

4

Problems(4)