MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
1995 Korea National Olympiad
1995 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(6)
Day 2
1
Hide problems
x^{n+2}+ax^{n+1}+bx^{n}+a+b cannot be decomposed
Let
a
,
b
a,b
a
,
b
be integers and
p
p
p
be a prime number such that: (i)
p
p
p
is the greatest common divisor of
a
a
a
and
b
b
b
; (ii)
p
2
p^2
p
2
divides
a
a
a
. Prove that the polynomial
x
n
+
2
+
a
x
n
+
1
+
b
x
n
+
a
+
b
x^{n+2}+ax^{n+1}+bx^{n}+a+b
x
n
+
2
+
a
x
n
+
1
+
b
x
n
+
a
+
b
cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than
1
1
1
.
Day 1
1
Hide problems
area of triangle conctructed by 3 perpendiculars on sides ABC
Let
O
O
O
and
R
R
R
be the circumcenter and circumradius of a triangle
A
B
C
ABC
A
BC
, and let
P
P
P
be any point in the plane of the triangle. The perpendiculars
P
A
1
,
P
B
1
,
P
C
1
PA_1,PB_1,PC_1
P
A
1
,
P
B
1
,
P
C
1
are drawn from
P
P
P
on
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
. Express
S
A
1
B
1
C
1
/
S
A
B
C
S_{A_1B_1C_1}/S_{ABC}
S
A
1
B
1
C
1
/
S
A
BC
in terms of
R
R
R
and
d
=
O
P
d = OP
d
=
OP
, where
S
X
Y
Z
S_{XYZ}
S
X
Y
Z
is the area of
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
.
Day 3
1
Hide problems
smallest number locks & total number of keys to open a box
Let
m
,
n
m,n
m
,
n
be positive integers with
1
≤
n
<
m
1 \le n < m
1
≤
n
<
m
. A box is locked with several padlocks which must all be opened to open the box, and which all have different keys. The keys are distributed among
m
m
m
people. Suppose that among these people, no
n
n
n
can open the box, but any
n
+
1
n+1
n
+
1
can open it. Find the smallest possible number
l
l
l
of locks and then the total number of keys for which this is possible.
Problem 3
1
Hide problems
maximum product of inradii by a cevian in equilateral
Let
A
B
C
ABC
A
BC
be an equilateral triangle of side
1
1
1
,
D
D
D
be a point on
B
C
BC
BC
, and
r
1
,
r
2
r_1, r_2
r
1
,
r
2
be the inradii of triangles
A
B
D
ABD
A
B
D
and
A
D
C
ADC
A
D
C
. Express
r
1
r
2
r_1r_2
r
1
r
2
in terms of
p
=
B
D
p = BD
p
=
B
D
and find the maximum of
r
1
r
2
r_1r_2
r
1
r
2
.
Problem 1
1
Hide problems
Korea 1995
For any positive integer
m
m
m
,show that there exist integers
a
,
b
a,b
a
,
b
satisfying
∣
a
∣
≤
m
\left | a \right |\leq m
∣
a
∣
≤
m
,
∣
b
∣
≤
m
\left | b \right |\leq m
∣
b
∣
≤
m
,
0
<
a
+
b
2
≤
1
+
2
m
+
2
0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}
0
<
a
+
b
2
≤
m
+
2
1
+
2
Problem 2
1
Hide problems
integer function with squares
find all functions from the nonegative integers into themselves, such that:
2
f
(
m
2
+
n
2
)
=
f
2
(
m
)
+
f
2
(
n
)
2f(m^2+n^2)=f^2(m)+f^2(n)
2
f
(
m
2
+
n
2
)
=
f
2
(
m
)
+
f
2
(
n
)
and for
m
≥
n
m\geq n
m
≥
n
f
(
m
2
)
≥
f
(
n
2
)
f(m^2)\geq f(n^2)
f
(
m
2
)
≥
f
(
n
2
)
.