MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
2013 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
6
1
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Tangent circles
Let
O
O
O
be circumcenter of triangle
A
B
C
ABC
A
BC
. For a point
P
P
P
on segmet
B
C
BC
BC
, the circle passing through
P
,
B
P, B
P
,
B
and tangent to line
A
B
AB
A
B
and the circle passing through
P
,
C
P, C
P
,
C
and tangent to line
A
C
AC
A
C
meet at point
Q
(
≠
P
)
Q ( \ne P )
Q
(
=
P
)
. Let
D
,
E
D, E
D
,
E
be foot of perpendicular from
Q
Q
Q
to
A
B
,
A
C
AB, AC
A
B
,
A
C
. (
D
≠
B
,
E
≠
C
D \ne B, E \ne C
D
=
B
,
E
=
C
) Two lines
D
E
DE
D
E
and
B
C
BC
BC
meet at point
R
R
R
. Prove that
O
,
P
,
Q
O, P, Q
O
,
P
,
Q
are collinear if and only if
A
,
R
,
Q
A, R, Q
A
,
R
,
Q
are collinear.
2
1
Hide problems
3 variable inequality
Let
a
,
b
,
c
>
0
a, b, c>0
a
,
b
,
c
>
0
such that
a
b
+
b
c
+
c
a
=
3
ab+bc+ca=3
ab
+
b
c
+
c
a
=
3
. Prove that
∑
c
y
c
(
a
+
b
)
3
(
2
(
a
+
b
)
(
a
2
+
b
2
)
)
1
3
≥
12
\sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }
cyc
∑
(
2
(
a
+
b
)
(
a
2
+
b
2
))
3
1
(
a
+
b
)
3
≥
12
8
1
Hide problems
Two lines separating plane
For positive integer
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
there are
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
points on plane which none of three are collinear. Prove there exist two lines
l
1
,
l
2
l_1, l_2
l
1
,
l
2
such that (1)
l
1
,
l
2
l_1, l_2
l
1
,
l
2
are not parallel. (2)
l
1
,
l
2
l_1, l_2
l
1
,
l
2
do not pass through any of
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
points. (3) There are
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
points on each region separated by two lines
l
1
,
l
2
l_1, l_2
l
1
,
l
2
.
7
1
Hide problems
Integer sequences
For positive integer
k
k
k
, define integer sequence
{
b
n
}
,
{
c
n
}
\{ b_n \}, \{ c_n \}
{
b
n
}
,
{
c
n
}
as follows:
b
1
=
c
1
=
1
b_1 = c_1 = 1
b
1
=
c
1
=
1
b
2
n
=
k
b
2
n
−
1
+
(
k
−
1
)
c
2
n
−
1
,
c
2
n
=
b
2
n
−
1
+
c
2
n
−
1
b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1}
b
2
n
=
k
b
2
n
−
1
+
(
k
−
1
)
c
2
n
−
1
,
c
2
n
=
b
2
n
−
1
+
c
2
n
−
1
b
2
n
+
1
=
b
2
n
+
(
k
−
1
)
c
2
n
,
c
2
n
+
1
=
b
2
n
+
k
c
2
n
b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n}
b
2
n
+
1
=
b
2
n
+
(
k
−
1
)
c
2
n
,
c
2
n
+
1
=
b
2
n
+
k
c
2
n
Let
a
k
=
b
2014
a_k = b_{2014}
a
k
=
b
2014
. Find the value of
∑
k
=
1
100
(
a
k
−
a
k
2
−
1
)
1
2014
\sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }
k
=
1
∑
100
(
a
k
−
a
k
2
−
1
)
2014
1
5
1
Hide problems
Functional Equation with gcd and lcm
Find all functions
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
satisfying
f
(
m
n
)
=
lcm
(
m
,
n
)
⋅
gcd
(
f
(
m
)
,
f
(
n
)
)
f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) )
f
(
mn
)
=
lcm
(
m
,
n
)
⋅
g
cd
(
f
(
m
)
,
f
(
n
))
for all positive integer
m
,
n
m,n
m
,
n
.
4
1
Hide problems
Integer sequence with fibonacci formula
{
a
n
}
\{a_n\}
{
a
n
}
is a positive integer sequence such that
a
i
+
2
=
a
i
+
1
+
a
i
(
i
≥
1
)
a_{i+2} = a_{i+1} + a_{i} (i \ge 1)
a
i
+
2
=
a
i
+
1
+
a
i
(
i
≥
1
)
. For positive integer
n
n
n
, define
{
b
n
}
\{b_n\}
{
b
n
}
as
b
n
=
1
a
2
n
+
1
∑
i
=
1
4
n
−
2
a
i
b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i }
b
n
=
a
2
n
+
1
1
i
=
1
∑
4
n
−
2
a
i
Prove that
b
n
b_n
b
n
is positive integer, and find the general form of
b
n
b_n
b
n
.
3
1
Hide problems
Existence of polynomial
Prove that there exist monic polynomial
f
(
x
)
f(x)
f
(
x
)
with degree of 6 and having integer coefficients such that (1) For all integer
m
m
m
,
f
(
m
)
≠
0
f(m) \ne 0
f
(
m
)
=
0
. (2) For all positive odd integer
n
n
n
, there exist positive integer
k
k
k
such that
f
(
k
)
f(k)
f
(
k
)
is divided by
n
n
n
.
1
1
Hide problems
Circumcenters
Let
P
P
P
be a point on segment
B
C
BC
BC
.
Q
,
R
Q, R
Q
,
R
are points on
A
C
,
A
B
AC, AB
A
C
,
A
B
such that
P
Q
∥
A
B
PQ \parallel AB
PQ
∥
A
B
and
P
R
∥
A
C
PR \parallel AC
PR
∥
A
C
.
O
,
O
1
,
O
2
O, O_{1}, O_{2}
O
,
O
1
,
O
2
are the circumcenters of triangle
A
B
C
,
B
P
R
,
P
C
Q
ABC, BPR, PCQ
A
BC
,
BPR
,
PCQ
. The circumcircles of
B
P
R
,
P
C
Q
BPR, PCQ
BPR
,
PCQ
meet at point
K
(
≠
P
)
K (\ne P)
K
(
=
P
)
. Prove that
O
O
1
=
K
O
2
OO_{1} = KO_{2}
O
O
1
=
K
O
2
.