MathDB

A quadrilateral and some Simson and Euler lines

Source: Romania TST 1 2012, Problem 2

5/3/2012

Let

ABCD

be a cyclic quadrilateral such that the triangles

BCD

and

CDA

are not equilateral. Prove that if the Simson line of

A

with respect to

\triangle BCD

is perpendicular to the Euler line of

BCD

, then the Simson line of

B

with respect to

\triangle ACD

is perpendicular to the Euler line of

\triangle ACD

.

Eulergeometrycircumcircleparallelogramgeometric transformationreflectioncyclic quadrilateral

Tangents to circle concurrent on a line

Source: Romania TST 3 2012, Problem 2

5/11/2012

Let

\gamma

be a circle and

l

a line in its plane. Let

K

be a point on

l

, located outside of

\gamma

. Let

KA

and

KB

be the tangents from

K

to

\gamma

, where

A

and

B

are distinct points on

\gamma

. Let

P

and

Q

be two points on

\gamma

. Lines

PA

and

PB

intersect line

l

in two points

R

and respectively

S

. Lines

QR

and

QS

intersect the second time circle

\gamma

in points

C

and

D

. Prove that the tangents from

C

and

D

to

\gamma

are concurrent on line

l

.

projective geometrygeometry proposedgeometry

Circumscribed quadrilateral and equal sums of angles

Source: Romania TST 2 2012, Problem 2

5/10/2012

Let

ABCD

be a convex circumscribed quadrilateral such that

\angle ABC+\angle ADC<180^{\circ}

and

\angle ABD+\angle ACB=\angle ACD+\angle ADB

. Prove that one of the diagonals of quadrilateral

ABCD

passes through the other diagonals midpoint.

geometrycircumcircletrigonometrysymmetrygeometric transformationreflectionrectangle

Find a sum

Source: Romania TST 5 2012, Problem 2

5/17/2012

Let

n

be a positive integer. Find the value of the following sum

\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},

where

e_k\in\{0,1\}

for

1\leq k \leq n

, and the sum

\sum_{(n)}

is taken over all

2^n

possible choices of

e_1,\ldots ,e_n

.

geometric seriesalgebra proposedalgebra

2

Problems(4)