MathDB

Fixed point

Source: Moldova TST 2006 Test II problem 2

3/25/2006

Let

C_1

be a circle inside the circle

C_2

and let

P

in the interior of

C_1

,

Q

in the exterior of

C_2

. One draws variable lines

l_i

through

P

, not passing through

Q

. Let

l_i

intersect

C_1

in

A_i,B_i

, and let the circumcircle of

QA_iB_i

intersect

C_2

in

M_i,N_i

. Show that all lines

M_i,N_i

are concurrent.

geometrycircumcirclepower of a pointradical axisgeometry proposed

Right-angled triangle and maximal area

Source: Moldavian TST_1, Problem 2

3/6/2006

Consider a right-angled triangle

ABC

with the hypothenuse

AB=1

. The bisector of

\angle{ACB}

cuts the medians

BE

and

AF

at

P

and

M

, respectively. If

{AF}\cap{BE}=\{P\}

, determine the maximum value of the area of

\triangle{MNP}

.

geometryanalytic geometrytrigonometrygeometry proposed

Modlova 3rd tst, problem 2

Source: Moldova TST III

3/26/2006

Let

n\in N

n\geq2

and the set

X

with

n+1

elements. The ordered sequences

(a_{1}, a_{2},\ldots,a_{n})

and

(b_{1},b_{2},\ldots b_{n})

of distinct elements of

X

are said to be

<span class='latex-italic'>separated</span>

if there exists

i\neq j

such that

a_{i}=b_{j}

. Determine the maximal number of ordered sequences of

n

elements from

X

such that any two of them are

<span class='latex-italic'>separated</span>

. Note: ordered means that, for example

(1,2,3)\neq(2,3,1)

.

combinatorics proposedcombinatorics

2

Problems(3)