MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
1994 Korea National Olympiad
1994 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(3)
Problem 3
2
Hide problems
identities with triangle of excenters
In a triangle
A
B
C
ABC
A
BC
,
I
I
I
and
O
O
O
are the incenter and circumcenter respectively,
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
the excenters, and
O
′
O'
O
′
the circumcenter of
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
. If
R
R
R
and
R
′
R'
R
′
are the circumradii of triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
, respectively, prove that: (i)
R
′
=
2
R
R'= 2R
R
′
=
2
R
(ii)
I
O
′
=
2
I
O
IO' = 2IO
I
O
′
=
2
I
O
cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25
Let
α
,
β
,
γ
\alpha,\beta ,\gamma
α
,
β
,
γ
be the angles of
△
A
B
C
\triangle ABC
△
A
BC
. a) Show that
c
o
s
2
α
+
c
o
s
2
β
+
c
o
s
2
γ
=
1
−
2
c
o
s
α
c
o
s
β
c
o
s
γ
cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma
co
s
2
α
+
co
s
2
β
+
co
s
2
γ
=
1
−
2
cos
α
cos
β
cos
γ
. b) Given that
c
o
s
α
:
c
o
s
β
:
c
o
s
γ
=
39
:
33
:
25
cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25
cos
α
:
cos
β
:
cos
γ
=
39
:
33
:
25
, find
s
i
n
α
:
s
i
n
β
:
s
i
n
γ
sin\alpha : sin\beta : sin\gamma
s
in
α
:
s
in
β
:
s
inγ
.
Problem 2
2
Hide problems
csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12
Let
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
be the angles of a triangle. Prove that
c
s
c
2
α
2
+
c
s
c
2
β
2
+
c
s
c
2
γ
2
≥
12
csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12
cs
c
2
2
α
+
cs
c
2
2
β
+
cs
c
2
2
γ
≥
12
and find the conditions for equality.
S_1 = {1}, S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}
Given a set
S
⊂
N
S \subset N
S
⊂
N
and a positive integer n, let
S
⊕
{
n
}
=
{
s
+
n
/
s
∈
S
}
S\oplus \{n\} = \{s+n / s \in S\}
S
⊕
{
n
}
=
{
s
+
n
/
s
∈
S
}
. The sequence
S
k
S_k
S
k
of sets is defined inductively as follows:
S
1
=
1
S_1 = {1}
S
1
=
1
,
S
k
=
(
S
k
−
1
⊕
{
k
}
)
∪
{
2
k
−
1
}
S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}
S
k
=
(
S
k
−
1
⊕
{
k
})
∪
{
2
k
−
1
}
for
k
=
2
,
3
,
4
,
.
.
.
k = 2,3,4, ...
k
=
2
,
3
,
4
,
...
(a) Determine
N
−
∪
k
=
1
∞
S
k
N - \cup _{k=1}^{\infty} S_k
N
−
∪
k
=
1
∞
S
k
. (b) Find all
n
n
n
for which
1994
∈
S
n
1994 \in S_n
1994
∈
S
n
.
Problem 1
2
Hide problems
integer functions
Let
S
S
S
be the set of nonnegative integers. Find all functions
f
,
g
,
h
:
S
→
S
f,g,h: S\rightarrow S
f
,
g
,
h
:
S
→
S
such that f(m\plus{}n)\equal{}g(m)\plus{}h(n), for all
m
,
n
∈
S
m,n\in S
m
,
n
∈
S
, and g(1)\equal{}h(1)\equal{}1.
integer equation
Consider the equation y^2\minus{}k\equal{}x^3, where
k
k
k
is an integer. Prove that the equation cannot have five integer solutions of the form (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4). Also show that if it has the first four of these pairs as solutions, then 63|k\minus{}17.