2005 Korea National Olympiad

Part of Korea National Olympiad

Subcontests

(8)

1

19th kmo #8

A group of 6 students decided to make study groups and service activity groups according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.

1

19th kmo #7

For a positive integer

n

f(n)

be the number of factors of

n^2+n+1

. Show that there are infinitely many integers

n

f(n) \geq f(n+1)

1

19th kmo #6

x_1, x_2, x_3, \cdots , x_n

x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1

\frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} .

1

19th kmo #5

P

be a point that lies outside of circle

O

. A line passes through

P

and meets the circle at

A

B

, and another line passes through

P

and meets the circle at

C

D

A

P

B

C

P

D

. Let the intersection of segment

AD

BC

L

E

(PA

BL \cdot PE = DL \cdot PD

M

is the midpoint of the segment

DE

M

is the intersection of lines

PL

DE

1

19th kmo #4

f: \mathbb R \to\mathbb R

such that for all real numbers

x

f(x) \geq 0

and for all real numbers

x

y

f(x+y)+f(x-y)-2f(x)-2y^2=0.

1

19th kmo #3

For a positive integer

K

, define a sequence,

\{a_n\}_n

a_1=K

, a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, for all

n\geq 1

. Find the smallest value of

K

a_{2005}

the first term equal to 0.

1

19th kmo #2

ABC

P

Q

\angle BPA + \angle AQC=90^{\circ}

. It is provided that the vertices of the triangle

BAP

ACQ

are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be

N

A \neq N

A

is the only intersection

A=N

), and the midpoint of segment

BC

M

. Show that the length of

MN

does not depend on

P

Q

1

19th kmo #1

For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of

a+b

\frac{a^{2005}+b^{2005}}{a+b}