2013中国女子数学奥林匹克试题及其解答(5)

Let the tangent Line

of

the line

Let

.

at .

at intersects with the

line

at and intersects with

. We are to consider the questions as below in the range of

(1) Let denote the area of triangles

of

such that

(2) Let

Note :

Let

. by , respectively. Find the range be the domain enclosed by line segments

contains line segments

and . and . be the maximum area of the triangle with a vertex

and draw the graph, then find the exterme value. which is contained in .

Find the function

Note : A

function has local minimum (or local maximum) at a

point

, which means for all points which is closed to

,

holds. We call local maximum, local minimum as extreme

value.

2.证明(Luis Gonzá

lez)Let

of

touching

at

and

Clearly

but since

and

Let be the incircle

are homothethic with incircles

( is

are symmetric about the midpoint of and

the M-excircle of MAB), it follows that

with corresponding cevians

2.证明

(Andrew64)As shown in the figure below.

is the intersection of

and

.

are homothetic