2013 APMO

Part of APMO

Subcontests

(5)

1

Another quadrilateral in a circle

ABCD

be a quadrilateral inscribed in a circle

\omega

P

be a point on the extension of

AC

PB

PD

\omega

. The tangent at

C

PD

Q

AD

R

E

be the second point of intersection between

AQ

\omega

B

E

R

1

Sets of integers with i+a in A or i-b in B

a

b

be positive integers, and let

A

B

be finite sets of integers satisfying (i)

A

B

are disjoint; (ii) if an integer

i

belongs to either to

A

B

i+a

A

i-b

B

a\left\lvert A \right\rvert = b \left\lvert B \right\rvert

\left\lvert X \right\rvert

denotes the number of elements in the set

X

1

Sequence of floors is arithmetic progression

2k

a_1, a_2, ..., a_k

b_1, b_2, ..., b_k

define a sequence of numbers

X_n

by X_n = \sum_{i=1}^k [a_in + b_i] (n=1,2,...). If the sequence

X_N

forms an arithmetic progression, show that

\textstyle\sum_{i=1}^k a_i

must be an integer. Here

[r]

denotes the greatest integer less than or equal to

r

1

Floor of square root

Determine all positive integers

n

\dfrac{n^2+1}{[\sqrt{n}]^2+2}

is an integer. Here

[r]

denotes the greatest integer less than or equal to

r

1

Triangles with Equal Areas

ABC

be an acute triangle with altitudes

AD

BE

CF

O

be the center of its circumcircle. Show that the segments

OA

OF

OB

OD

OC

OE

dissect the triangle

ABC

into three pairs of triangles that have equal areas.