MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
1997 Korea National Olympiad
1997 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
8
1
Hide problems
Algebra like NT
For any positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
and
w
,
w,
w
,
prove that
x
2
,
y
2
,
z
2
x^2,y^2,z^2
x
2
,
y
2
,
z
2
and
w
2
w^2
w
2
cannot be four consecutive terms of arithmetic sequence.
7
1
Hide problems
Jacobi wasn't well known back in 1997
Let
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
be the points outside the
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
B
A
Z
=
∠
C
A
Y
,
∠
C
B
X
=
∠
A
B
Z
,
∠
A
C
Y
=
∠
B
C
X
.
\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.
∠
B
A
Z
=
∠
C
A
Y
,
∠
CBX
=
∠
A
BZ
,
∠
A
C
Y
=
∠
BCX
.
Prove that the lines
A
X
,
B
Y
,
C
Z
AX, BY, CZ
A
X
,
B
Y
,
CZ
are concurrent.
6
1
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Polynomial Problem
Find all polynomial
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
for any reals
x
,
y
x,y
x
,
y
such that (i)
x
100
+
y
100
≤
P
(
x
,
y
)
≤
101
(
x
100
+
y
100
)
x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})
x
100
+
y
100
≤
P
(
x
,
y
)
≤
101
(
x
100
+
y
100
)
(ii)
(
x
−
y
)
P
(
x
,
y
)
=
(
x
−
1
)
P
(
x
,
1
)
+
(
1
−
y
)
P
(
1
,
y
)
.
(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).
(
x
−
y
)
P
(
x
,
y
)
=
(
x
−
1
)
P
(
x
,
1
)
+
(
1
−
y
)
P
(
1
,
y
)
.
5
1
Hide problems
Ineq in Geo again
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the side lengths of any triangle
△
A
B
C
\triangle ABC
△
A
BC
opposite to
A
,
B
A,B
A
,
B
and
C
,
C,
C
,
respectively. Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be the length of medians from
A
,
B
A,B
A
,
B
and
C
,
C,
C
,
respectively. If
T
T
T
is the area of
△
A
B
C
\triangle ABC
△
A
BC
, prove that
a
2
x
+
b
2
y
+
c
2
z
≥
3
T
\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}
x
a
2
+
y
b
2
+
z
c
2
≥
3
T
4
1
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Classic Number Theory
For any prime number
p
>
2
,
p>2,
p
>
2
,
and an integer
a
a
a
and
b
,
b,
b
,
if
1
+
1
2
3
+
1
3
3
+
⋯
+
1
(
p
−
1
)
3
=
a
b
,
1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},
1
+
2
3
1
+
3
3
1
+
⋯
+
(
p
−
1
)
3
1
=
b
a
,
prove that
a
a
a
is divisible by
p
.
p.
p
.
3
1
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Ineq in Geo
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon such that
A
B
=
B
C
,
C
D
=
D
E
,
E
F
=
F
A
.
AB=BC,CD=DE, EF=FA.
A
B
=
BC
,
C
D
=
D
E
,
EF
=
F
A
.
Prove that
B
C
B
E
+
D
E
D
A
+
F
A
F
C
≥
3
2
\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}
BE
BC
+
D
A
D
E
+
FC
F
A
≥
2
3
and find when equality holds.
2
1
Hide problems
Classic Sum
For positive integer
n
,
n,
n
,
let
a
n
=
∑
k
=
0
[
n
2
]
(
n
−
2
k
)
(
−
1
4
)
k
.
a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.
a
n
=
∑
k
=
0
[
2
n
]
(
k
n
−
2
)
(
−
4
1
)
k
.
Find
a
1997
.
a_{1997}.
a
1997
.
(For real
x
,
x,
x
,
[
x
]
[x]
[
x
]
is defined as largest integer that does not exceeds
x
.
x.
x
.
)
1
1
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Classic Old Combo
Let
f
(
n
)
f(n)
f
(
n
)
be the number of ways to express positive integer
n
n
n
as a sum of positive odd integers. Compute
f
(
n
)
.
f(n).
f
(
n
)
.
(If the order of odd numbers are different, then it is considered as different expression.)