2008 APMO

Part of APMO

Subcontests

(5)

1

a problem with residues

a, b, c

be integers satisfying 0 < a < c \minus{} 1 and

1 < b < c

k

0\leq k \leq a

r_k,0 \leq r_k < c

be the remainder of

kb

when divided by

c

. Prove that the two sets

\{r_0, r_1, r_2, \cdots , r_a\}

\{0, 1, 2, \cdots , a\}

1

Function with nonnegative integers

Consider the function

f: \mathbb{N}_0\to\mathbb{N}_0

\mathbb{N}_0

is the set of all non-negative integers, defined by the following conditions :

(i)

f(0) \equal{} 0;

(ii)

f(2n) \equal{} 2f(n) and

(iii)

f(2n \plus{} 1) \equal{} n \plus{} 2f(n) for all

n\geq 0

.

(a)

Determine the three sets L \equal{} \{ n | f(n) < f(n \plus{} 1) \}, E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}, and G \equal{} \{n | f(n) > f(n \plus{} 1) \}.

(b)

k \geq 0

, find a formula for a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\} in terms of

k

1

Groups of Students

Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is

46

, there is a set of

10

students in which no group is properly contained.

1

Two lines meet at circle

\Gamma

be the circumcircle of a triangle

ABC

. A circle passing through points

A

C

meets the sides

BC

BA

D

E

, respectively. The lines

AD

CE

\Gamma

G

H

, respectively. The tangent lines of

\Gamma

A

C

DE

L

M

, respectively. Prove that the lines

LH

MG

\Gamma

1

prove that BPC<120

ABC

be a triangle with

\angle A < 60^\circ

X

Y

be the points on the sides

AB

AC

, respectively, such that CA \plus{} AX \equal{} CB \plus{} BX and BA \plus{} AY \equal{} BC \plus{} CY . Let

P

be the point in the plane such that the lines

PX

PY

are perpendicular to

AB

AC

, respectively. Prove that

\angle BPC < 120^\circ