1

Part of 2013 China National Olympiad

Problems(2)

CA, AP and PE are the side lengths of a right triangle

Source: 2013 China Mathematical Olympaid P1

K_1

K_2

of different radii intersect at two points

A

B

C

D

be two points on

K_1

K_2

, respectively, such that

A

is the midpoint of the segment

CD

. The extension of

DB

K_1

at another point

E

, the extension of

CB

K_2

at another point

F

l_1

l_2

be the perpendicular bisectors of

CD

EF

, respectively. i) Show that

l_1

l_2

have a unique common point (denoted by

P

). ii) Prove that the lengths of

CA

AP

PE

are the side lengths of a right triangle.

geometrycircumcircleperpendicular bisectorpower of a pointgeometry proposed

Find the minimum

Source: 2013 China Mathematical Olympaid P4

n \geqslant 2

be an integer. There are

n

{A_1},{A_2},\ldots,{A_n}

which satisfy the condition \left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \forall i,j \in \left\{ {1,2,...,n} \right\}. Find the minimum of

\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}

floor functioninductioninequalitiescombinatorics proposedcombinatorics